U.S. patent application number 11/174800 was filed with the patent office on 2005-12-01 for patterning technology for folded sheet structures.
This patent application is currently assigned to Rutgers, The State University of New Jersey. Invention is credited to Kling, Daniel H..
Application Number | 20050267616 11/174800 |
Document ID | / |
Family ID | 26925971 |
Filed Date | 2005-12-01 |
United States Patent
Application |
20050267616 |
Kind Code |
A1 |
Kling, Daniel H. |
December 1, 2005 |
Patterning technology for folded sheet structures
Abstract
The present invention supplies practical procedures, functions
or techniques for folding tessellations. Several tessellation
crease pattern techniques, and the three-dimensional folded
configuration are given. Additionally several new forming
processes, including mathematical methods for describing the
material flow are discloseddoubly-periodic folding of materials
that name the doubly-periodic folded (DPF) surface, including
vertices, edges, and facets, at any stage of the folding. This
information is necessary for designing tooling and forming
equipment, for analyzing strength and deflections of the DPFs under
a variety of conditions, for modeling the physical properties of
DPF laminations and composite structures, for understanding the
acoustic or other wave absorption/diffusion/reflection
characteristics, and for analyzing and optimizing the structure of
DPFs in any other physical situation. Fundamental methods and
procedures for designing and generating DPF materials include ways
for defining the tessellation crease patterns, the folding process,
and the three-dimensional folded configuration. The ways are
mathematically sound in that they can be extended to a
theorem/proof format.
Inventors: |
Kling, Daniel H.; (Ringoes,
NJ) |
Correspondence
Address: |
KLAUBER & JACKSON
411 HACKENSACK AVENUE
HACKENSACK
NJ
07601
|
Assignee: |
Rutgers, The State University of
New Jersey
|
Family ID: |
26925971 |
Appl. No.: |
11/174800 |
Filed: |
July 5, 2005 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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11174800 |
Jul 5, 2005 |
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09952057 |
Sep 14, 2001 |
|
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6935997 |
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60232416 |
Sep 14, 2000 |
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Current U.S.
Class: |
700/98 |
Current CPC
Class: |
G06T 17/20 20130101 |
Class at
Publication: |
700/098 |
International
Class: |
G06F 019/00 |
Claims
1. A method for providing a pattern for folded sheet structures
comprising: (a) entering row and column data and an intersection
point thereof into a computer program; (b) calculating a
doubly-periodic folded (DPF) surface pattern from the data entered
in step (a); (c) outputting the DPF surface pattern calculated in
step (b) for folding a sheet structure according to said DPF
surface pattern.
2. The method according to claim 1, wherein step (c) includes
outputting the DPF surface pattern to a folding machine.
3. The method according to claim 2, wherein the folding machine
receiving the output in step (c) is a casting machine.
4. The method according to claim 2, wherein the folding machine
receiving the output in step (c) is a cutting machine.
5. The method according to claim 4, wherein the cutting machine is
a milling machine.
6. The method according to claim 2, wherein the folding machine
receiving the output in step (c) is a stamping machine.
7. The method according to claim 1, wherein the DPF surface pattern
is output to a display.
8. The method according to claim 1, wherein the DPF surface pattern
is printed.
9. The method according to claim 2, wherein the DPF surface pattern
is output to a computer numerical control (CNC) machine.
10. The method according to claim 1, wherein the row data entered
in step (a) comprises a row of edges in a tessellation (RET).
11. The method according to claim 1, wherein the row data entered
in step (a) comprises a row cross section (RCS).
12. The method according to claim 1, wherein the row data entered
in step (a) comprises a row of edges (RED) of a folded sheet.
13. The method according to claim 10, wherein the row of edges all
have a same fold convexity and are coplanar.
14. The method according to claim 12, the row of edges all have a
same fold convexity and are coplanar.
15. The method according to claim 1, wherein step (a) includes at
least one vertex of the row data and column data.
16. The method according to claim 1, wherein the row data entered
in step (a) comprises incremental vectors in polar coordinates.
17. The method according to claim 1, wherein the column data
entered in step (a) comprises a column of edges in a tessellation,
augmented to give relative amplitudes and spacing of successive
rows of the tessellation (CET).
18. The method according to claim 10, wherein the column data
entered in step (a) comprises a column of edges in a tessellation,
augmented to give relative amplitudes and spacing of successive
rows of the tessellation (CET).
19. The method according to claim 12, wherein the column data
entered in step (a) comprises a column of edges in a tessellation,
augmented to give relative amplitudes and spacing of successive
rows of the tessellation (CET).
20. The method according to claim 17, wherein the column of edges
alternate sequentially in fold convexity and are coplanar in a
vertically oriented plane.
21. The method according to claim 18, wherein the column of edges
alternate sequentially in fold convexity and are coplanar in a
vertically oriented plane.
22. The method according to claim 19, wherein the column of edges
alternate sequentially in fold convexity and are coplanar in a
vertically oriented plane.
23. The method according to claim 1, wherein the column data
entered in step (a) comprises a column-cross section (CCS).
24. The method according to claim 1, wherein the column
data.entered in step (a) comprises a column strip map.
25. The method according to claim 10, wherein the column data
entered in step (a) comprises a column-cross section (CCS).
26. The method according to claim 12, wherein the column data
entered in step (a) comprises a column strip map.
27. The method according to claim 1, wherein the row data is
entered in step (a) for a plurality of rows.
28. The method according to claim 1, wherein the column data is
entered in step (a) for a plurality of columns.
29. The method according to claim 27, wherein steps (a) and (b) are
repeated for each one of the plurality of rows, before step (c) is
performed only once.
30. The method according to claim 28, wherein steps (a) and (b) are
repeated fore each one of the plurality of rows, before step (c) is
performed only once.
31. The method according to claim 27, wherein for each one of the
plurality of rows, step (a), (b) and (c) are respectively
performed.
32. The mehtod according to claim 28, wherein for each one of the
plurality of columns, steps (a), (b) and (c) are respectively
performed.
33. The method according to claim 27, wherein the DPF surface
pattern step (b) includes at least one row of facets, said facets
being connected in a row and column direction so that said at least
one row of facets is bounded on either side by a row of edges, and
the facets are connected successively across column edges.
34. The method according to claim 33, wherein said at least one row
of facets comprises a plurality of rows of facets, wherein a
row-cross section being an intersection of a plane with a ruled
surface, each one the rows of facets being a family of parallel
line segments.
35. The method according to claim 28, wherein the plurality of
columns are parallel.
36. The method according to claim 1, wherein the calculating of the
DPF surface pattern begins with a general DPF pattern which is
adjusted according to the row and column data from step (a).
37. The method according to claim 11, wherein the RCS is based on
three dimensions comprising x, y and z dimensions.
38. The method according to claim 12, wherein the RED is based on
three dimensions comprising x,y and z dimensions.
39. The method according to claim 37, wherein the row data for the
RCS is supplied on the XZ plane, and the column data is supplied on
the YZ plane.
40. The method according to claim 38, wherein the row data for the
RED is supplied on the XZ plane, and the column data is supplied on
the YZ plane.
41. The method according to claim 10, wherein the RET is based on
three dimensions comprising x,y and z dimensions.
42. The method according to claim 41, wherein the row data for the
RET is supplied on the XY plane, and the column data on the YZ
plane.
43. The method according to claim 39, wherein an axis of the X
dimension is used as a reference line.
44-50. (canceled)
Description
[0001] This application is based on and claims priority from
provisional application 60/232,416 filed Sep. 14, 2000.
BACKGROUND OF THE INVENTION
[0002] 1. Field of the Invention
[0003] The present invention relates to folded tessellations and
other folded strucures. More particularly, the present invention
relates to specific configurations, and patterning methods,
applicable to the unfolded sheet, the three-dimensional folded
structure, proccesses of transforming a sheet to a folded strucure,
and machine descriptions for the same.
[0004] 2. Description of the Related Art
[0005] A well-known problem in the art of designing and forming
materials into folded networks is that with the exception of
deformation at the fold, the material is not signifgantly streched,
and this imposes simultaneous constraints on the vertices, edges,
and facets of a proposed structure and on the process of forming
such a structure. However folded structures have many advantages
over structures produced by other means such as casting, stamping
or assembling, such as cost of manufacture and the versatility to
many sheet materials.
SUMMARY OF THE INVENTION
[0006] Accordingly, the present invention discloses our methods and
procedures for designing and generating folding networks and
tessellations. Several means for defining the crease patterns, and
the three-dimensional folded configuration are given. Additionally
several new forming processes, including mathematical methods for
describing the material flow are disclosed. The description and use
of practical methods, data structures, functions and techniques
that name the folded surface, including vertices, edges, and
facets, and that describe the forming processes for sheet material
is disclosed. This information is valuable for designing tooling
and forming equipment, for analyzing strength and deflections of
the DPFs under a variety of conditions, for modeling the physical
properties of DPF laminations and composite structures, for
understanding the acoustic or other wave
absorption/diffusion/reflection characteristics, and for analyzing
and optimizing the structure of DPFs in any other physical
situation. Many other aspects of our folding technology are also
presented.
BRIEF DESCRIPTION OF THE DRAWINGS
[0007] FIG. 1 illustrates the creasing along a tessellation to
produce a PDF.
[0008] FIG. 2 illustrates the column and row phenomenon.
[0009] FIG. 3 is a chart comparing various folding methods
according to the present invention.
[0010] FIG. 4 illustrates arrays of DPFs generated by using Row
Cross Section (RCS) and Colum Cross Section (CCS).
[0011] FIG. 5 illustrates Z-values of RCS are the offsets on the
CCS.
[0012] FIG. 6 shows a simple folding tessellation according to the
present invention.
[0013] FIG. 7 illustrates steps (a) through (d) in the
Wave-Tessellation Method.
[0014] FIG. 8 illustrates fold convexity at a Vertex.
[0015] FIG. 9 illustrates fold locations on a strip.
[0016] FIG. 10 illustrates the defining of a strip map by polygon
correspondence.
[0017] FIG. 11 illustrates several strip maps according to the
present invention.
[0018] FIG. 12 illustrates the extending of a strip map to
construct DPFs.
[0019] FIG. 13 shows vertex calculations with strip-maps to
construct DPFs.
[0020] FIG. 14 is a flowchart of the procedure for determining the
correspondence function from an Unfolded Sheet to a Folded Sheet
according to an embodiment of the present invention.
[0021] FIG. 15 illustrates compositions of Local Isometries.
[0022] FIG. 16 illustrates the geometrical comparison of DPF-Vertex
Calculations.
[0023] FIG. 17 illustrates the trigonometric relationships between
entry data.
[0024] FIG. 18 illustrates two reflection patterns for a line.
[0025] FIG. 19 illustrates a local Isometry applied to two
surfaces.
[0026] FIG. 20 illustrates that almost all the components cannot
fold gradually.
[0027] FIG. 21 illustrates the conventional prior art creasing
process.
[0028] FIG. 22 illustrates the novel creasing process according to
the present invention.
[0029] FIG. 23 shows a comparison of the conventional creasing
process versus the novel creasing process according to the present
invention.
[0030] FIG. 24 illustrates the implementation of a two phase
forming process using rollers to fold.
[0031] FIG. 25 illustrates parameterized RED and CCS data for a
Chevron Pattern application.
[0032] FIG. 26 illustrates vertex coordinates with DPF pattern type
parameters.
[0033] FIG. 27 provides a table illustrating coordinates for the
vertex of a chevron pattern using the wave-fold method and
parameterized RED and CCS according to the present invention.
[0034] FIG. 28 illustrates a plurality of wave patterns w1-19
according to the present invention.
[0035] FIG. 29 illustrates another plurality of wave patterns
w10-w17 according to the present invention.
[0036] FIG. 30 illustrates yet another plurality of wave patterns
w18-w26 according to the present invention.
[0037] FIG. 31 illustrates a machine implementing a process
according to the present invention.
[0038] FIG. 32 illustrates five different frames with parameters
overhead.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0039] (Introduction and Definitions)
[0040] It should be understood by persons of ordinary skill in the
art that the methods and procedures described here in Part I refer
to folding an idealized infinitely thin plane. The plane is
deformed, without changing any intrinsic lengths along the surface
itself, to produce multi-faceted geometric structures. Any surface
folded from a plane will have zero-curvature. This means the
Jacobian of the Gauss map is zero on smooth regions, that the sum
of the geodesic curvatures along an edge singularity (crease), when
measured from the two sides of the edge, will total zero, and that
at each vertex the cone angle will total 2.pi.=360 degrees. The
zero-curvature results from the fact that the material is not
stretched by folding, and so the surface remains isometric to the
plane.
[0041] In practice a sheet has finite thickness and may have
intentional or unintentional distortions in superposition to the
ideal process described here. The idealized zero-curvature surface
is the architectural base, from which close to zero-curvature
surfaces will be designed. For instance, even if a folding pattern
for sheet material is designed identically to the folding pattern
for a plane, the physical surface will not have zero-curvature
perfectly, because the thickness of the sheet causes a bend radius
at each fold, and generally this forces very small regions of
positive curvature and negative curvature near the vertices.
Moreover it may be desirable to select an idealized folding pattern
to plan the major displacement of the sheet material, and then
exploit the plasticity of the material to make minor adjustments to
the surface configuration. The design difficulties and
manufacturing advantages for these surfaces close to zero-curvature
surfaces are very much the same as for zero-curvature surfaces, and
so this is a valuable application of our technology. Typically
close to zero-curvature surfaces and zero-curvature surfaces differ
by a minor perturbation. Thus by using the idealized zero-curvature
surface as an architectural base, these valuable close to
zero-curvature surfaces are both theoretically and in practice an
application of the our technology. However the key step is
determining the idealized architectural base, and this is the focus
of our numerical and geometrical methods.
[0042] The technology disclosed here also applies to repetitive
folding patterns. These idealized surfaces may have one or two
directions of repetition. The repetitions may correspond to
translation or rotational symmetry. A doubly-periodic surface with
zero curvature is called a folded tessellation. In practice, folded
tessellations are finite and may have intentional or unintentional
imperfections and so may have no exact translational or rotational
symmetries.
[0043] In this disclosure the term DPF refers to the patterned
surfaces generated by our methods and technology presented here.
The idealized DPF will have exactly zero curvature. Moreover, by
entering non-periodic data into the procedures and methodology,
many valuable non-periodic folding patterns can be designed by DPF
technology. By entering periodic data into our methodologies, the
resulting surfaces will be folded tessellations. A physical folded
sheet is called a DPF if its design has an architectural base from
an idealized DPF, although the scope of DPF technology is much
broader, including sheet material with regions or components that
are designed in part by DPF procedures and methodology, with or
without intentional or unintentional variations into close to zero
curvature geometry.
[0044] (In reference to other documents, the meaning of the
accronym DPF has gone through a series of changes, and at one time
refered abstractly to doubly-periodic flat surfaces that may or may
not have existed in only three-dimensions.)
[0045] Each of our various designing methods has its own
generalizations for defining DPFs. The main class of DPFs are the
doubly-periodic folding structures describable by all of our
designing methods. The simplest class of DPFs is included in the
main class and has a more obvious array structure and also the
straightforward two translation symmetries, and will be described
first. Until otherwise stated, the discussion in part I will refer
to DPFs in the simplest class. The terms and techniques given are
part of the invention, and generally will not apply to folding
structures produced outside our technology.
[0046] FIG. 1 shows four stages in the Uniform DPF Process for
producing a typical DPF in the simplest class. In the Uniform DPF
Process, the individual facets are assumed to be rigid and only the
fold angles of the edges may change. In our Uniform process the DPF
flexes with one parameter of motion. Data from any one stage will
determine the entire process. In FIG. 1A-D this parameter, called
the flex parameter, varies from zero to its maximum .pi./2=90
Degrees.
[0047] Notice the edges form a net (also called a mesh or a graph)
on the surface, with degree four vertices and quadrilateral
regions. By analogy with the usual rectangular grid, this enables
the edges to be divided into two types, according to whether they
advance in the row or column direction. In one direction (for the
simplest class) the edges form parallel lines in the fold crease
tessellation, as shown in FIG. 1A. The end-to-end chains in this
direction are considered to be the columns of edges (left to right
in FIGS. 1& 2). The other edges form end-to-end chains that are
considered to be the rows of edges (top to bottom in the figures).
In FIG. 2, A an exemplary row and column are shown in boldface
lines.
[0048] It can be seen that at any stage of the folding process, the
edges in any given row all have the same fold convexity and are
coplanar, and that the edges in any given column alternate
sequentially in fold convexity and are coplanar in a vertically
oriented plane.
[0049] Two other observations will be necessary concerning cross
sections in the column and row directions of a folded DPF. Of
course, the facets connect in the row and column direction to form
chains called a row and column offacets. A row of facets will be
bounded on either side by a row of edges, and will have its facets
connected successively across individual column edges. At any stage
of folding all of the column edges within a row of facets are
parallel. Furthermore, an entire row of facets can be developed
into a family of line segments parallel to its column edges, as
shown in FIG. 2C. Viewing a row of facets as a family of parallel
line segments, and extending the line segments if necessary to a
larger ruled surface, one chooses a plane normal to the parallel
lines of the ruled surface and defines the row cross-section to be
the intersection of the plane with the ruled surface. Each row of
facets will be seen to have the same row cross section--although if
one uses the orientations available the row cross section
alternates in mirror image with each subsequent row of facets.
[0050] In the fold crease tessellation of an unfolded DPF, the two
columns of edges that bound a column of facets on both sides form
parallel lines. During the Uniform DPF Process, the vertical planes
that contain the columns of edges are all parallel. Moreover note
any other line drawn on the tessellation parallel to the column
edges will also lie in a vertical plane throughout the process. The
plane will be in fact parallel to the planes containing the columns
of edges. This folded drawn line is called a column cross-section.
Several column cross-sections are drawn on one column of edges in
the DPF in FIG. 2D.
[0051] It is sufficient (for the simplest class still) to specify
the equivalent of one row and one column and their intersection
point to determine the entire DPF surface. Our procedures and
methods differ in how the row and column data is implied
mathematically.
[0052] As shown in FIG. 3, for the row data, three presentations
are currently available:
[0053] 1. A row of edges in the tessellation (RET).
[0054] 2. The row cross-section (RCS).
[0055] 3. A row of edges on the folded DPF (RED).
[0056] As these are piecewise-linear coplanar curves, it suffices
to enter the vertices on a plane for one unit of repetition with
the number of repetitions desired. This is the prefered data entry
in our computer programs, although other methods, such as
describing the edges as incremental vectors in polar coordinates,
have also been used.
[0057] In addition, as shown in FIG. 3, for the column data, three
presentations are currently available:
[0058] 1. A column of edges in the tessellation, augmented to give
the relative amplitudes and spacing of successive rows of the
tessellation (CET).
[0059] 2. The column cross-section (CCS).
[0060] 3. A column strip map (CSM).
[0061] For the first two it again suffices to enter the
two-dimensional vertices for one unit of repetition and the number
of repetitions desired. A similar reduction is possible for the
strip map and is discussed, infra, with the Strip-Map Method in
with regard to strip map design and shown in FIGS. 9-13.
[0062] FIG. 3 summarizes our procedures and methods by their
row/column data requirements, and gives an overview of their
respective advantages.
[0063] As coordinates become necessary x, y and z will denote the
row, column, and vertical directions respectively. The row data RCS
and RED will be supplied on an XZ plane, RET on an XY plane, and
the column data supplied on an YZ plane. The RCS and RED Z
coordinates are rotated relative to the three dimensional z
coordinate.
[0064] With exception of tessellation data, the row and column data
are usually assumed to refer to the to the DPF in its intended
position. For the Wave-Tessellation Method, the flex parameter must
be given additionally in some form. Each method also has the option
for changing the intended flex parameter if desired, and can
implement the change within the Method by adjusting the data
entered to fit the new stage of folding as described, infra.
[0065] For the main class of DPFs it is also possible to translate
between methods, as described in Section 9. Thus the designer may
visualize the DPF from several perspectives and create a DPF to
meet variously stated requirements. Moreover, while editing,
rescaling, and partially folding or unfolding, the user may jump
back and forth between methods. This flexibility enables the data
to be developed in the terms of the methods most convenient for the
application, so one can find and optimize the choice of DPF
readily.
[0066] The methods share some common features relating to the
format of the data entered and the data out-putted. To generate
folded tessellations, first for efficiency one may enter just one
unit of the row and column data along with their respective
repetition numbers. One may apply the repetitions to the entry data
to generate the full row and column information and then apply the
procedures to calculate the DPF surface, or one may input into the
procedure one row unit and one column unit, apply the procedure,
and then apply the repetitions in the row and column direction to
generated the full surface. In some cases, such as when the row and
column data do not have a standard repetition, one chooses to use
the full row and column data for the entry data.
[0067] Also the piecewise linear (PL) structure of the entry data
and the out-putted DPF surface means that they may be linearly
interpolated from their vertices, and the interpolation can occur
before or after the application of the procedures. For instance
with the Two Cross-Section Method, for each pair of segments, with
one from RCS and one from CCS, the method will generate a
quadrilateral facet on the DPF. The method takes the two endpoints
of the two segments to the four vertices of the quadrilateral.
However, the convex-hull of the four output vertices defines the
quadrilateral. Thus it may be computationally more efficient to use
just the vertices of the PL curves to generate the vertices of the
facets of the DPF and then interpolate to construct the facet
regions, then to use the fully connected PL curves to generate the
facet regions. The methodology to exploit this fully for these
folding maps is described in Section 7.
[0068] Each of the computer codes for the methods also have
rescaling subroutines and other convenient techniques for adjusting
the elements in the data entered. For instance often one wishes to
experiment within a DPF pattern type. This implies there is a type
for both the row data and column data, defined by restricting
certain of their symmetries and other characteristics. Within the
restrictions, one may parameterize the possibilities allowed for
the row and column data types and use this as a front end for our
computational procedures. The user then experiments efficiently
with these parameters, which generate the row and column data,
which in turn generates the DPF within the selected pattern type.
This is further discussed in Section 11.
[0069] 3 Two Cross Section Method
[0070] 3.1 Overview
[0071] This method generates DPFs directly in three-space and has
various generalizations. In the simplest form, the user enters a
row and column cross sections by simply naming their vertices on a
coordinate plane. The cross-sections can be designed effortlessly.
The resulting infinite table contains DPFs with great commercial
application. Moreover the direct relationship between the data
entered and the resulting structure will enable engineers to design
materials with custom performance properties. For both chains a
variety of front ends are possible, such as entering a single unit
of repetition may be entered along with the number of repetitions
desired in each direction, or using a computer code to generate the
chains within selected parametric conditions. The designer
typically selects a general DPF pattern type from the infinite
array for its structural characteristics, and then optimizes the
parameters in that pattern type by adjusting the parameters in the
corresponding RCS and CCS wave types. Even by changing just the
proportions within the two cross-sections, the qualities of the DPF
can change widely.
[0072] 3.2 Technical Procedure
[0073] This method is perhaps the easiest to use. The user enters
the row cross-section (RCS) and the column cross-section (CCS) onto
an XZ-plane and a YZ-plane, respectively. Optionally the row and
column repetition numbers may be factored out, and the data entered
may be represented by the two-dimensional vertices of the PL curves
as described previously. The x-axis of the RCS is used as a
reference line, and the RCS given should have no undercuts. In
Figure REF[DPFchart], `A,B,C` show three row cross sections and the
`1,2,3` show three column cross sections. For `C` the RCS is a sine
wave (not a PL curve) and so the DPFs generated are outside the
simplest class. Note for pictorial quality the data is not to scale
and the RCS and CCS are shown to two repetitions, while the DPFs
are shown to at least three repetitions in each direction.
[0074] The idea is to position each facet-row in three-space, and
then attach them along their common edge-rows. -RCS will denote the
reflection of RCS about its x-axis, and RCS and -RCS will denote
the usual corrugation-type surfaces formed by extending RCS and
-RCS from the XZ-plane to three-space in the direction parallel to
the y-axis.
[0075] The reference plane of these two surface is then the XY
plane. Facet-rows will be taken from RCS and -RCS alternately, and
positioned sequentially according to the segments of CCS. To do
this label the consecutive edges of CCS with +1 and -1 alternately.
The YZ plane containing CCS should be considered as usual to lie in
the XYZ space. On each segment of CCS labeled +1, position a copy
of RCS so that its reference plane contains the segment, its
reference plane is normal to the YZ plane, and has not shifted in
the x direction. Likewise position -RCS on the remaining edges of
CCS labeled with -1.
[0076] Corresponding and parallel to the original series of edges
in CCS, we now have a series of corrugation surfaces. Each of these
consecutive surfaces should be cropped along the curve where it
intersects the two adjacent corrugation surfaces in the series.
These cropped corrugation surfaces then splice together to make the
DPF.
[0077] Alternatively, RCS and CCS may be used to construct the DPF
by generating the collection of column cross-sections of the DPF
obtained by intersecting it with each plane parallel to the
YZ-plane. To determine the shape of these column cross-sections,
consider a point (x0, z0) on the RCS. The column cross-section in
the {x0} XRXR plane will be the same as the original CCS, but
offset according to z0. To do this the segments L of CCS are
assigned a plus/minus value as above. For segment L on CCS, a new
L' is drawn parallel to L and at a distance of
.vertline.z0.vertline. from L, with L' above or below L according
to the product of the signs of z0 and L. To finish this column
cross section in the {x0}.times.R.times.R plane, the various L'
produced from the sequence of segments in CCS are extended or
cropped to join their ends sequentially. FIG. 5 shows an
example.
[0078] To generate just the vertices of the DPF, the column
cross-section need only be calculated for points (X,Z) that are
vertices of RCS. The calculations for the Method may be computed in
a number of ways. Suppose (X.sub.i,Z.sub.i) is a vertex of RCS, and
L.sub.j-1 and L.sub.j are two consecutive edges of CCS. The value
Z.sub.i represents how the vertex is offset from its centerline.
Consider the line segment L.sub.j-1 in the YZ plane, and let
L.sub.j-1' be the line parallel to the segment and offset in the YZ
plane be Zi. L.sub.j-1' should be offset above or below depending
on the product of the sign of Zi and the +/-label of the segment.
Similarly define L.sub.j' offset from the line segment L.sub.j
Calculate the point of intersection
(y.sub.j,z.sub.j)=L.sub.j-1'.andgate.- L.sub.j'. Then
(X.sub.i,y.sub.j,z.sub.j) is a vertex of the DPF. This is shown in
the third frame of FIG. 5.
[0079] In summary the x coordinate of the vertices requires no
calculations and the y and z coordinates are calculated by parallel
and intersecting line formulas. The fourth frame of FIG. 5 shows
the ith column of edges found in the X.sub.i.times.R.times.R
plane.
[0080] 4 Wave-Tessellation Method
[0081] 4.1 Overview
[0082] The Wave-Tessellation Method described here produces the
planar tessellations on the sheet material that that specifies
where the edge creases will be on the folded sheet. Additionally
the convexity of each crease is also specified. Of course for a
generic tessellation to be the crease patterns is impossible, and
in general it is quite difficult to design the fold crease
tessellation without a system or design methodology. The fold
crease tessellation is usually given independently of the final
structure in three-space in which case additional information must
be supplied. Knowing the fold crease tessellation is essential as a
"parts list" showing the size and connection arrangement of the
facets. The Wave Tessellation Method may also be most suited for
studying the flexural properties of a DPF under mechanical
forces.
[0083] Tessellations outside DPF technology will most often have no
or perhaps isolated three-dimensional positions possible.
Tessellations generated by our Wave-Tessellation Method will have
infinitely many three-dimensional positions, enabling the DPF to be
produced simply, by stringing together these positions
continuously, to give the Uniform DPF Process, as described in
Section 16. An explanation of the failure of generic tessellations
outside our Wave-Tessellation Method is also given there.
[0084] 4.2 Constructing the Tessellation
[0085] The tessellation method is remarkably simple. Lines are
drawn in the XY-plane parallel to the y axis and with periodic
spacing. These will become the columns of edges. An example is
shown in FIG. 7A. A piecewise-linear curve with one vertex on each
column-line is drawn to serve as one of the rows of edges in the
tessellation (RET) as shown in FIG. 7B. This curve is only required
to project onto the x axis injectively, and should be periodic,
although in practice it generally will have slope bounded within
[-3,3]. Multiple copies of the row are placed on the plane by
translating it in the y-direction. Additionally the row may be
dilated in the y-direction with either positive or negative
factors, with translates in the y-direction of the re-scaled rows
placed on the plane. FIG. 7C shows scaling factors to the left of
the rows.
[0086] For DPFs in the main class, the various copies of the row
must not cross and and they should have a periodic pattern in the
y-direction. These minimal constraints produce a tessellation that
if given the fold convexities described below, will fold into a
DPF. For DPFs in the simplest class, the rows must not intersect
and a balanced convexity condition must be met. These variations
are discussed in Section 16.
[0087] The name `Wave Tessellation` refers to the fact that the
tessellation is constructed by essentially drawing multiple copies
of the same `waves` (RET) shifted and with various amplitudes.
[0088] 4.3 Fold Convexity Specification
[0089] Since the edges in the tessellation are unfolded, a system
of assigning fold convexities to the edges must be known in advance
of the folding process. For DPFs designed by the Tessellation
Method above, our system described below will satisfy the internal
constraints of the sheet as a linkage and yield one parameter of
motion. For general tessellations outside our methodology, other
complex systems of assigning fold convexity may be necessary if the
sheet can be folded at all.
[0090] The reader may follow the discussion on the Method for
assigning the fold convexity to the edges of the tessellation along
with FIG. 7D in which the solid lines become convex folds and the
dotted lines become concave folds. The edges within any single row
will all have the same convexity. The convexity of the initial row
is selected (convex in the figure). For any two neighboring rows
consider the ratio of their scaling factors. If the ratio is
positive the neighboring rows have opposite convexities, if the
ratio is negative then the neighboring rows are given like
convexities. This enables all of the rows to be given a convexity
assignment.
[0091] Next the convexity of each column edge can be calculated
locally by examining the vertex at either of its endpoints and
knowing the row convexity at that vertex. A schematic of the
possible relationships between a column edge convexity and the row
edge convexity is shown in FIG. 8, with the solid and dashed lines
indicating convex and concave folds, respectively. At each vertex
the row (generally horizontal in the figure) will divide the
surface into two angles that total 360 degees. The column edge that
is on the larger side of the row should have the same convexity as
the row, and the column edge on the smaller side should have the
opposite convexity. This implies the convexities of the edges in
the column chains alternate each time they cross a row. As each
column edge has a convexity that can be determined from either of
its endpoints, there is a redundancy to this Method that can be
shown to specify the fold convexities consistently.
[0092] 4.4 Computation Formula
[0093] For computer entry data, select an initial wave
W.sub.0:R.fwdarw.R, seen as the graph of a function in the
XY-plane, positioned so W.sub.0(0)=0. This is RET, and for PL waves
W.sub.0 may be specified succinctly by naming its vertices (xy).
The position of wave W.sub.n may then be defined inductively. Each
wave W.sub.n has a pair of (y.sub.n,a.sub.n), where y.sub.n is the
incremental spacing on the y-axis, y.sub.n=W.sub.n(0)-W.sub.n-1(0),
and an is the amplitude for W.sub.n relative to W.sub.0. The
sequence {(y.sub.1,a.sub.1), (y.sub.2,a.sub.2), . . . } is called
CET, usually written explicitly only up to one cycle of repetition.
Conversely from CET and RET one constructs W.sub.n by
Wn(x):=an Wo(x)+Wn-1(0)+yn=an Wo(x)+y1+y2+ . . . yn
[0094] 5 Wave-Fold Method
[0095] This method is one of the most intuitive. The terminology
emphasizes that edges in a row all have the same fold convexity by
viewing each row of edges as a single "wave fold". The user enters
either a row of edges on the tessellation (RET) or a row of edges
on the folded DPF (RED), and enters the column cross-section (CCS)
in the folded DPF. Depending upon whether RET or RED is used for
the row data, the method can proceed along two different ways.
[0096] 5.1 General Description of Method
[0097] Assume first that RED data is given. One of the potential
application for using RED is that it can be designed as the contact
area where a DPF core material meets the laminated faces, and
customizing this junction has structural and gluing ramifications.
As mentioned RED is a coplanar PL curve and can be specified by
two-dimensional vertices in some AZ plane. The point where RED and
CCS intersect must be specified in the data. The point will be a
vertex of CCS. For convenience, the placement of RED on the
XZ-plane is given with the x-axis passing through this point. The x
axis will be used as a reference line for RED. The PL curve CCS is
given by its two-dimensional vertices in the YZ plane. The Method
proceeds by considering each vertex of CCS, its perpendicular
bisector, and the plane in XYZ space normal to the bisector and
containing the vertex. In XYZ space each of these planes is
parallel to the x-axis, and tangent to its defining vertex of CCS.
On each of these planes a copy of RED is drawn such that the
reference line of RED is parallel to the x axis and passes through
the vertex of CCS that defined the plane. The copies of RED are all
positioned on each of these planes so that their x coordinates of
RED have not changed.
[0098] The next task is to adjust the amplitudes of the copies of
RED. The re-scaling will occur within the plane containing the
copy, with the restriction that the reference line and the x
coordinates remain fixed. The remaning axis of the plane will be
re-scaled linearly with potentially a positive or negative
coefficient.
[0099] The copy of RED lying in the plane tangent to the vertex of
CCS specified as the intersection vertex is not re-scaled. From
here the copies of RED are re-scaled consecutively, following the
sequencing of the vertices of CCS that were used to construct their
planes. Consider two consecutive vertices of CCS, and the
corresponding consecutive copies of RED. We may assume the first
copy of RED is scaled properly as the Method proceeds inductively.
Select a vertex of the first copy of RED that is not on its
centerline. Draw a segment from this vertex to the corresponding
vertex of the second RED. Note these corresponding vertices have
the same x coordinate. Thus the second RED can be re-scaled so that
this segment is parallel to the segment in the YZ plane joining the
considered vertices of CCS. This reduces the computations
explicitly to simple plane manipulations and slope calculations.
Section 5.2 below carries out this Method explicitly without using
an inductive procedure to find the scaling factor of RED and the
(x,y,z) coordinates of an arbitrary vertex of the DPF.
[0100] Instead of using the row data expressed as RED, it may be
more convenient to express it as a row of edges in the
tessellation, RET. This could be valuable if one wants to control
the angles and x dimensions of the individual facets in the
constructed DPF.
[0101] One proceeds similarly with the planes tangent to the
vertices of CCS and perpendicular to their bisectors. One then
determines the folded shape of RET at the vertex of CCS. This shape
is of course RED, and so the Method can then proceed as explained
using RED. Section 5.3 explains the conversion process from RET to
RED.
[0102] 5.2 Method for a Vertex
[0103] This section demonstrates the Method explicitly, and finds
the scaling factor of RED and the (xyz) coordinates of the vertices
of the DPF. To carry out the computations, call the ith vertex of
RED (X.sub.i,Z.sub.i) and the jth vertex of CCS (y.sub.j,z.sub.j);
we will calculate the (i,j)th vertex of the DPF. To simplify
notation, for any vector u, <u>=u/.vertline.u.vertline..
Define the edge vector leading up to (y.sub.j,z.sub.j) by
v.sub.j=(y.sub.j-y.sub.j-1,z.sub.j-z.s- ub.j-1). At each vertex
(y.sub.j,z.sub.j) of CCS, define y.sub.j to be one half the
included angle, and b.sub.j to be the unit bisecting vector.
Then
.gamma..sub.j=arccos(<-v.sub.j>.multidot.<v.sub.j+1>)/2b.sub.j-
=v.sub.j+1-v.sub.j
[0104] Each copy of RED must be adjusted by an amplitude factor at
each vertex of CCS. Assuming the specified intersection point of
CCS and RED is given as (y.sub.1,z.sub.1) on CCS and as X=0 on RED,
the amplitude factor k.sub.j for RED at the vertex
(y.sub.j,z.sub.j) of CCS is
k.sub.j=(-1){circumflex over (
)}(j+1)cos(.gamma..sub.1)/cos(.gamma..sub.j- )
[0105] The (i,j) vertex of the DPF is calculated to be (x,y,z)
below
x=X.sub.i
y=y.sub.j+k.sub.jZ.sub.ib.sub.jz
z=z.sub.j-k.sub.jZ.sub.ib.sub.jy
[0106] Since there is no included angle on the first and last
vertices of CCS, some convention
[0107] such as having CCS start with (y0, z0) and not calculating
RED on the ends of CCS is necessary.
[0108] 5.3 Method Using RET
[0109] When the Wave-Fold Method uses RET for row data, it is easy
to convert RET to RED and then apply either the recursive procedure
in Section 5.1 or the formulation in Section5.2. We assume again
that RET and CCS intersect at (X.sub.0,Y.sub.0)=(0,0) on RET and
(y.sub.1,z.sub.1) on CCS with one half the included angle at
.gamma..sub.j located at (y.sub.j,z.sub.j) on CCS.
[0110] RED will consist of a sequence of segments corresponding
exactly in length to those on RET, but with their slopes adjusted.
For segment (X.sub.i-1,Y.sub.i-1), (X.sub.i,Y.sub.i) its angle of
ascent is
.alpha..sub.i=arctan(Y.sub.i-1-Y.sub.i)/X.sub.i-1-X.sub.i)) and
length l.sub.i={square root}((X.sub.i-1-X.sub.i){circumflex over (
)}.sub.2+(Y.sub.i-1-Y.sub.i){circumflex over ( )}.sub.2). The
corresponding segment on RED will have angle of ascent
.delta..sub.i=arcsin(sin(.alpha..sub.i)/sin(.gamma..sub.i))
[0111] The vertex (X.sub.i,Z.sub.i) on RED can be defined
recursively as
(X.sub.0,Y.sub.0)=(0,0)
(X.sub.i,Z.sub.i)=(X.sub.i-1+l.sub.i
cos(.delta..sub.i),Z.sub.i-1+l.sub.i sin(.delta..sub.i))
[0112] These equations enable one to convert RET to RED. Additional
formula are in Section 9
[0113] 6 Strip-Map Method
[0114] The Strip Map Method is a unique among the DPF procedures in
the way its column data (a column strip-map, CSM) and its row data
(row cross-section, RCS) emphasize certain three-dimensional
properties of a DPF. While defining a strip map to be used for the
column data requires somewhat awkward data and a sub-Method of its
own, there may be some advantages of using the Strip Map Method
that out-weigh the added difficulty: This method gives the greatest
control over the silhouette and space surrounding the DPF; It has
significant flexibility for generating DPFs outside the main class;
And it displays natural limitations on the entry data for assuring
the DPF is embedded. Before describing how the strip map is applied
to produce the DPF, our strip map representation and sub-method
will be explained. The terminology Strip Map Procedure will refer
to the sub-method that determines the strip map from entry data and
Strip Map Method will refer to the larger procedure that utilizes
strip maps to produce DPFs.
[0115] A strip map can be explained heuristically quite easily, as
a way of folding a rectangular piece of paper to lay flush on a
plane. The actual computations however, typically require
techniques using re-iterative compositions, piecewise defined local
isometries of the plane, linear interpolations, and/or local
coordinate systems defined on each facet. Furthermore, the strip
and the strip map should be represented in data structure and
methodology to interface well with designing strip maps and
application of the strip map to the Strip Map Method to design DPF
surface configurations. Additionally, the strip map should be
reduced to numerical parameters that enable convenient manipulation
for both the data entry of the strip map and the application of the
strip map to DPF design. We have developed several procedures for
representing, calculating, and designing strip maps that solves
these difficulties. Before addressing the mechanism of our
procedures, some formal definitions are given.
[0116] A local isometry f: R{circumflex over (
)}n.fwdarw.R{circumflex over ( )}m, n<=m is a piecewise smooth
map that preserves arc length. That is if for all rectifiable
.gamma.:[a,b].fwdarw.M
.intg..vertline.f(.gamma.(s))'.vertline.ds=.intg..sub.ab.vertline..gamma.(-
s)'.vertline.ds.
[0117] For f: G.fwdarw.H with GR{circumflex over ( )}n and
HR{circumflex over ( )}m the definition applies also. A strip is a
rectangle {(y,z).vertline..gamma..epsilon.J, z.epsilon.K}, where J
and K are intervals. In practice K is usually many times shorter
then J. A strip map is a local isometry from a strip to the
plane.
[0118] 6.1 Strip Map Procedure 1
[0119] This section discloses our first strip map Method, the one
used to generate the images. It produces a continuous function from
the strip to the plane, by applying the folds to the strip
consecutively. In FIG. 9, the fold locations on the strip are shown
as segments (p.sub.i, q.sub.i). F.sub.i will denote folding the
strip along segment (p.sub.i, q.sub.i), keeping regions A.sub.j,
j<=i fixed. The strip map will then be the composition
F.sub.1.multidot.F.sub.2.multidot.F.sub.3.multidot.F.sub.4:st-
rip.fwdarw.plane. As the points' locations change with each
successive folding, it becomes awkward to determine whether they
should be left fixed or reflected by a map F.sub.i unless some
additional data is attached to the points identifying in which
regions they originated. For instance, use l=l(y,z) as the
subscript l of the region A.sub.l containing (y,z): It is computed
easily by simultaneous linear inequalities. Let R.sub.i be the
reflection across line (p.sub.i,q.sub.i). Then define the function
F.sub.i on labeled points (y,z,l) by 1 Fi ( y , z , l ) := ( y , z
, l ) if l <= i ( Ri ( y , z ) , l ) Otherwise
[0120] The strip map is then given using the composition
F:=F.sub.1.multidot.F.sub.2.multidot. . . . .multidot.F.sub.n-1
[0121] Note that for even complicated strip maps with interior fold
vertices, the Method will define the strip map by letting A.sub.1 .
. . A.sub.n be a path of regions connecting through the strip, and
R.sub.i being the reflection across the lowest edge of A.sub.i+1
used in the path. A rooted tree may be used instead of a path,
where "<" becomes the partial ordering of the tree.
[0122] 6.2 Strip Map Procedure 2
[0123] A strip map is singular on the fold creases and fold
vertices. This Method uses the singular set to apply a polygonal
structure to the strip before and after folding, with the edges and
vertices of the polygonalization coinciding with the singularities
or boundary, and the regions being the largest non-singular regions
of the strip. These two polygonal structures are entered as data,
and formatted so that the parts of strip polygonalization before
folding are easily matched with the corresponding parts after
folding in the image polygonalization. This added structure greatly
simplifies the calculations within the Method.
[0124] There are many data structures for representing polygonal
structures. For the strip in FIG. 9, it would suffice to enter an
array of vertices [[p.sub.0 . . . p.sub.n],[q.sub.0 . . . q.sub.n]]
for the strip and the array of vertices [[p.sub.o' . . .
p.sub.n'],[q.sub.0' . . . q.sub.n']] for the image of strip, where
the accent indicates the point's position in the folded strip. This
rectangular grid representation will work easily for any strip map
with vertices lying on the rectangles perimeter. In some cases
there may be triangular regions with two fold crease edges sharing
the same endpoint vertex; in this case the vertex can be listed
twice so that (p.sub.i,q.sub.i) is still an edge for all i. For
strip maps with interior vertices as in the last example of FIG. 11
a simple array will often be sufficient to describe the
polygonalization, but again it may be necessary to use some
redundancy in the listing of the vertices. More complicated data
structures can be used. One representation uses a set of lists of
vertices, where each list represents a polygon, by giving the
vertices in cyclic order around the polygons perimeter. In general
the data entered need not be any more complicated then giving the
vertices grouped in appropriate structure, for then with linear
inequalities it is possible to determine the edges and
polygons.
[0125] Once both the polygonalization of the unfolded and folded
strip are entered as data with formatting to reveal their
corresponding parts, the strip map is then determined by piecing
together isometries that are defined locally on each polygon:
Suppose (y,z) is a point on the strip lying in polygon A.sub.i, and
A.sub.i' is the corresponding polygon in the folded strip. Since
A.sub.i, and A.sub.i' are congruent, there is an isometry of the
plane F.sub.i that sends the corresponding parts of A.sub.i to
A.sub.i'. In particular F.sub.i(y,z) is (y',z'). To determine
F.sub.i, select three vertices u,v,w of A.sub.i that are
non-collinear. Their image u',v',w' in A.sub.i' are found without
calculation, by using the correspondence between the compatible
data structure of A.sub.i, and A.sub.i'. Using the information that
F.sub.i sends u,v,w to u',v',w' respectively, several geometric
methods are given below for determining F.sub.i. Thus by
representing the strip before and after folding in corresponding
formats to display the vertices in a described polygonal structure,
all of the F.sub.i can be determined and hence the entire strip
map. This key method has many applications to folding analysis and
is described briefly below and again in Section 7.
[0126] Local coordinate systems on A.sub.i and A.sub.i' can be
constructed so that corresponding points have the same
parameterization. For instance barycentric coordinates on u,v,w are
used in one of our computer codes, and then the parameters are used
to reproduce the correct point on the folded strip by using the
same barycentric coordinates on u',v',w'. The method in another of
our computer codes uses two vertices u,v, and determines the unit
vector n:=(v-u)/.vertline.v-u.vertline.; rotates n to find the
perpendicular m; and uses this as the two basis vectors for a local
orthonormal coordinate system with origin u. Likewise a local
orthonormal coordinate system with origin u' is determined. The
orientation of m' may be supplied by either a two coloring on the
polygons of the strip or by including the corresponding vertices w
on A.sub.i and w' on A.sub.i'. The displacement vector s:=(y,z)-u
is then converted using dot products in terms of the new
coordinates by s.sub.1:=n.multidot.s, an s.sub.2:=m.multidot.s. The
point (y',z') is calculated to be u'+sin'+s.sub.2m'. This local
coordinate systems method works well, and are layed out more
explicitly in Section 7. Symbolically we get F.sub.i(y,z)=(y',z'),
where each polygon pair A.sub.i,A.sub.i' results in different
F.sub.i.
[0127] The total strip map F is based on the individual F.sub.i. If
(y,z) lies in A.sub.i, this will insure F.sub.i(y,z) lies in
A.sub.i', and is the correct point (y',z'). Thus to determine the
effect of F on a point (y,z) on the strip, first determine which
region A.sub.j contains (y,z), and then apply F.sub.j. This is
shown schematically if FIG. 0. The various F.sub.i may be
calculated in advance before the Method receives the input (y,z),
or each entry (y,z) may be used to find the vertices to apply to
determining the function F.sub.i.
[0128] An efficient variation of this procedure is to give a
polygonalization of the unfolded strip, a selection of three
vertices of each polygon, and the position of the selected vertices
on the folded strip. This is essentially the same as above, but the
polygonalization of the folded strip has been reduced to selected
vertex correspondence. One proceeds as above with the local
coordinate systems on each polygon, with base points the three
selected vertices. The image of a point in a polygon is then
determined using the same coordinates, but with base points the
corresponding three vertices of the folded strip. Greater detail is
described in Section 7.
[0129] 6.3 Strip Map Procedure 3
[0130] This is our most recent Method for representing and
calculating a strip map. The idea is that the entry data can simply
be the polygonalization of the strip before folding; the Method can
then calculate the polygonalization of the strip after folding and
represent the folded strip in compatible format with the unfolded
strip; and then proceed using isometries, local coordinate systems,
or linear interpolation defined piecewise on the individual
polygons as above in Strip Map Method 2. This reduces the entry
data to just one polygonal structure, so that making design changes
in the strip map is simplified.
[0131] We have several sub-procedures to construct the folded
polygonalization from the unfolded polygonalization.
[0132] The Strip Method 1 can be applied to the vertices of the
unfolded polygonalization, to get the vertices of the folded
polygonalization, with the edge and polygon structure induced by
correspondence.
[0133] Following a connection path or rooted tree out through the
adjacency graph of the polygonalization (the dual of the
polygonalization graph), the successive polygons can be attached
alternating their orientations. One way to do this uses simple dot
products.
[0134] The polygons may be two-colored. Polygons of one color are
then all reflected into their mirror image. The polygons are then
reassembled, reconnecting along previous shared edges. To connect
the polygons, some procedure such as a path or tree through the
adjacency graph of the polygons should be performed, providing a
sequence for connecting the polygons.
[0135] A rooted planar tree on the edge graph is selected. Here
planar means at each vertex the edges in the tree are presented in
a list successively (for instance counter-clockwise), with the
angle included between successive edges of the list know and
containing no edges omitted from the tree. Using this information,
there is a natural way to two-color the included angles so that it
could be extended to a two coloring of the polygonalization. By
knowing all the edge lengths, the included angles, and the position
of the root edge, one can readily reconstruct the position of the
vertices of the entire tree and hence the polygonalization. To find
the folded polygonalization, the procedure is to reverse the sign
of the included angles assigned one of the two colors, and then
reconstruct the vertices of the tree, and so the folded
polygonalization.
[0136] These sub-procedures are similar, and generally require
iterative or sequential calculations to `build` the folded
polygonalization. Their programming is simplified, by copying the
data format of the folded polygonalization from the data format of
the unfolded polygonalization and the steps are then only required
to change the vertex coordinates in the given polygonalization of
the strip. Once computed the calculations for the general point
(yz) are computed using Strip Method 2.
[0137] 6.4 Variations
[0138] Depending on implementation, these procedures can be used to
calculate the strip map on all points in the strip or for perhaps
programming efficiency the strip map's effect on just the vertices
of the strip. If the latter is employed one may then calculate the
map's effect on the edges and polygons by interpolation as
explained in Section 6.3 and Section 7, and thus construct the
entire strip map if necessary for application. FIG. 13 shows an
efficient method for calculating just the the coordinates needed by
the vertices of the DPF.
[0139] It is also possible to blend these procedures in various
combinations. For instance given a polygonalization of the unfolded
strip choose one base polygon to remain stationary under the
folding map. To determine the location of a point on the strip map
after folding, a path is selected from the point to the base
polygon. Follow this path the methods of Strip Map Procedure 3 may
be employed to determine the points destination. Another
combination applies the variation in the end of the Strip Map
Procedure 2 Section to Strip Map Procedure 3, so that instead of
the polygonaliztion of the folded strip, only three vertices on
each polygon are needed. One additional note, that reflections
F.sub.i in Strip Map Method 1 are up to rigid motion of the YZ
plane defined by (y,z).fwdarw.(y,.vertline.z.vertline.).
[0140] 6.5 Designing Strip Maps
[0141] To design a strip map, one may work experimentally with a
strip of paper, and then compute geometrically the coordinates of
the folds and vertices before folding and after folding, and then
apply Strip Map Method 2. Alternatively just the unfolded
coordinates may be calculated, and apply Strip Map Method 3. To
proceed without experimentation, there is a constraint on the
polygonalization of the unfolded strip: If the strip contains
internal vertices the vertex must have an even number of adjacent
edges and by adding and subtracting the angles around the vertex
alternately the sum angle must equal zero.
[0142] To produce periodic folded surfaces, the strip map f:
R.times.[0,1].fwdarw.R{circumflex over ( )}.sub.2 should be
periodic, so for some minimal p>0 there is a translation v so
f(x,t)+v=f(x+p,t) for all x.epsilon.R, t .epsilon. [0,1]. To assure
the folded structure has no interference problems, the strip map f
should also be injective in the first coordinate, that is for all
x,y .epsilon.R, t .epsilon.[0,1], f(x,t)< >f(y,t), when x<
>y. FIG. 11 symbolically illustrates four examples that satisfy
these periodic and injective properties, by showing the edges and
vertices of the strip before and after folding. To reduce the
pictures to numerical maps, one applies one of the strip map
procedures given above.
[0143] 6.6 Designing Surfaces by the Strip Map Method
[0144] A strip map is a function F from one two-dimensional region
to another, F(y,z)=(y',z'). It may be extended to a function by
G(x,y,z)=(x,y',z'). So has no effect on the x-coordinate and is
just F on the y and z-coordinates. The significance of G is that
when it is applied to a zero-curvature surface S it will produce a
zero-curvature surface G(S). This follows as G=I.times.F, where
I:R.fwdarw.R is the identity, and set products of local isometries
are again local isometrics.
[0145] FIG. 12 show how a strip map F extends to and is applied to
produce a DPF. In A) and A') of the figure, the strip in the
yz-plane is seen before and after applying F. Parts B) and B')
extend this to xyz-space, and the effect of on the rectangular box
may be seen. In Part C), a standard corrugation surface S, having
zero-curvature, is placed in the box. The Strip Map Method applies
G to S, to produce the DPF surface G(S).
[0146] The Strip Map Method uses F:strip.fwdarw.plane and a
zero-curvature surface S to produce a DPF. For DPFs in the main
class, the cross section of S taken in any plane parallel to the
xz-plane will give the same curve. This curve is the row
cross-section (RCS) of the DPF. Note the z-coordinate of RCS
composes with the z-coordinate of the strip map. The strip
I.times.J used for the domain of F must have height interval J
sufficient to contain the vertical variation of RCS. Also, the
plane of RCS is perpendicular to the interval I of the strip. With
these assumptions the strip map is called a column strip map (CSM)
and the Method uses RCS and CSM as entry data.
[0147] As described above, the strip map F is extended to a map
G:Box.fwdarw.R{circumflex over ( )}.sub.3. The corrugated surface S
is designed to fit in the box, and then G(S) is the DPF. As it is
easy to construct a map g: R{circumflex over ( )}.sub.2.fwdarw.S
presenting the corrugation, as an option one may define
G.multidot.g: R.sub.2 R.sub.3 to give the map from the unfolded
plane to the DPF in three space. Whether using G or G.multidot.g,
the method as described so far gives the DPF as a continuous image
of a local isometry function. This has been implemented in MAPLE
computer code.
[0148] Alternatively, it is perhaps computationally more efficient
and precise to determine the effect of the strip map on the
vertices of the DPF, along with the edge and polygonal structure.
If needed the full continuous maps G.multidot.g and G can be
deduced by isometric interpolation, as explained in Section
REF[maprep]. The surfaces R.sub.2 and S can be polygonalized on
their singular set, so that G.multidot.g: R{circumflex over (
)}.sub.2.fwdarw.DPF and G.vertline.S.fwdarw.DPF are presented by
defining the vertex correspondence. FIG. 13 demonstrates an
efficient Method for determining the vertices needed.
[0149] The strip map method offers an direct picture of the
position of the surface in three-space. This method is most
valuable for designing DPFs with surface portions arranged in close
tolerance without intersecting. To assure the DPF will embed in
R{circumflex over ( )}.sub.3, two sufficient conditions can be met.
First, the RCS curve .gamma.R.times.[0,1] should project
R.times.[0,1].fwdarw.R injectively. This is the usual condition.
Secondly the strip map F:R.times.[0,1].fwdarw.R{circumflex over (
)}.sub.2 should be injective on each line R.times.{t} for each t
.epsilon.[0,1] The embeddedness follows immediately from the
infectivity hypothesis on F and RCS. The surface generated will be
doubly periodic if both F and RCS have one direction of
periodicity.
[0150] Additionally, we comment that the strip map method is not
restricted to use rectangular strips as described here for
simplicity, as long as it is a local isometry. Also a variation
rotates the surface S about a vertical linebefore applying G.
[0151] 7 Representation for Folded Structures
[0152] In many cases it is useful to describe the function that
carries points on the unfolded sheet to their locations on the
folded sheet. Such a function can also be relate a structure on the
unfolded sheet with the corresponding structure on the folded
sheet. For instance, such a function can describe the
correspondence between the components (vertices, edges, polygons)
of the unfolded tessellation with the components of folded surface.
For fiber composites, the function may be extended to the tangent
space to relate the thread direction of the sheet before folding
with the thread direction after folding. Similarly for perforated
materials, the function extends to map the perforations in the
unfolded sheet to those on the folded sheet. We have developed a
simple method for representing such a function, in the case where
the folded sheet has linear facets or can be approximated by a
zero-curvature surface with linear facets.
[0153] The method is uniquely related to folded surfaces. Surfaces
that are produced by conventional manufacturing processes such as
stamping, casting, and forging and that do not have zero-curvature
can not be represented by this method. Conversely while there are
many ways to represent a general surface, this Method is uniquely
efficient in its exploitation of the arc-length-preserving
characteristic of folding maps, enabling the entire function to be
interpolated from a sparse data point correspondence, using local
isometries expressed through simple coordinate systems.
[0154] The unfolded sheet has a singular set consisting of vertices
and edges, as does the folded sheet. These give a polygonalization
on both surfaces. Suppose v.sub.i=(x.sub.i,y.sub.i,z.sub.i) is a
vertex (the z.sub.i may be omitted on the unfolded sheet). An edge
can then be described by listing its two endpoints. If
v.sub.1=(x.sub.1,y.sub.1,z.sub- .1) and
v.sub.2=(x.sub.2,y.sub.2,z.sub.2) are its two endpoints, the points
along the edge are described by (1-t)v.sub.1+tv.sub.2, where
0<=t<=1. As before z may be omitted for the unfolded sheet. A
polygon region can be described by listing its vertices cyclically,
P=(v.sub.1,v2, . . . . v.sub.n). To check if on the unfolded sheet,
the point
(x-x.sub.k)(y.sub.k+1-y.sub.k)-(y-y.sub.k)(x.sub.k+1-x.sub.k)<0
[0155] (x,y) lies in the polygon
((x.sub.1,y.sub.1),(x.sub.2,y.sub.2), . . . (x.sub.n,y.sub.n)) one
may determine if
[0156] for all 1<=k<=n
[0157] The three dimensional case can be handled similarly by first
rotating the facet to lie in the XY-plane or by using vector
operations. For a point p known to lie in a polygon region
(v.sub.1,v.sub.2, . . . v.sub.n) in two or three dimensions, we
have several methods to re-coordinatize p relative to its polygon
vertices. Three vertices u,v,w should be selected from
(v.sub.1,v.sub.2, . . . v.sub.n). In each of the examples below,
new coordinates a, b (and c) are determined by solving the given
equations.
[0158] 1. Barycentric coordinates: With a+b+c=1
p=au+bv+cw
[0159] 2. Linear coordinates:
p=v+a(u-v)+b(w-v)
[0160] 3. Orthonormal coordinates:
p=ar+bt
[0161] where r,s,t is the orthonormal basis given by
r-(u-v)/.vertline.u-v.vertline.,s=(u-v).times.(w-v)/(u-v).times.(w-v),t-r.-
times.s
[0162] The user should choose a convenient rule for determining
which vertices u,v,w of the polygon will be selected, and should
choose a coordinate system such as one of those suggested above.
Once done, the entire map is computed separately for each polygon P
by generating the coordinates a,b above for each point pEP and then
using a,b to generate p' with exactly the same equations but
calculated using u',v',w' on the folded sheet, that correspond to
u, v, w.
[0163] The unique feature of this Method is that because it is
describing a folding map that preserves arc-length it can exploit
the fact that the distance (intrinsic and extrinsic) relationships
on each polygon expressing the points in terms of the vertices are
the same before and after folding. This enables a coordinate system
on a polygon of the unfolded sheet, with base points u, v, w, to
transfer identically to a coordinate system on a polygon of the
folded sheet, with base points u',v'w'. One way is to use isometric
interpolation along the edge and polygons to determine the full
surface. Effectively, the entire surface map is determined by the
correspondence on the vertices, between the unfolded and folded
sheet. This identical transfering of coordinate systems is quite
novel, because sheet material formed into similar structures by
state of the art methods can not be interpolated this way and
generally would require complicated non-linear relationships. FIG.
14 shows a flowchart of the steps of the procedure. In the figure,
x represents either a point p as above or a structure component on
the unfolded sheet such as thread direction, perforation location,
and others.
[0164] 8 Local Isometry Method
[0165] The Local Isometry Method is versatile for generating DPFs
and works with the zero-curvature properties of the DPF
transparently. It describes DPFs as the image of a local isometry
F:R{circumflex over ( )}.sub.2.fwdarw.R{circumflex over ( )}.sub.3.
Points on the unfolded sheet are assigned their folded locations in
three-space. As with the other methods, it may be advisable to
compute F just on the singular set or just on the vertices, and
then apply our isometric interpolation in Section 7 if the full map
is needed.
[0166] There are many combinations of ways to construct DPFs by
composing local isometries. In this section, we disclose data
structures and methods for representing and calculating these
functions. One word of caution however, in general it is difficult
to determine in advance if the abstract surfaces generated by this
method contain self-intersections, so some additional screening
(usually on a case by case basis) must applied to determine the
application to sheet materials. For simplicity in notation we will
describe the case where the DPF is an infinite sheet, with it
implied that for most applications the domain would be restricted
to a finite sheet.
[0167] 8.1 Background Examples
[0168] A folded sheet may be seen as the image of a local isometry
F:R{circumflex over ( )}.sub.2.fwdarw.R{circumflex over ( )}.sub.3.
Once produced F may be represented using the method of Section 7.
To design or create a local isometry F:R{circumflex over (
)}.sub.2.fwdarw.R{circumfle- x over ( )}.sub.3 is much more
difficult. The method in this section constructs local isometrics
F:R{circumflex over ( )}.sub.2.fwdarw.R{circu- mflex over (
)}.sub.3 by combining several simple local isometrics through set
products and composition. The method is illustrated with a first
example: Define :R.fwdarw.R by sending even integers to 0 and odd
integers to 1 and extending linearly:
(2Z)={0}, .fwdarw.(2Z+1)={1},
[0169] and d.vertline.(n,n+1)=.+-.1, where the sign corresponds to
the parity of n.epsilon.Z. is a local isometry that folds the line
onto the unit interval. Let I:R.fwdarw.R be the identity, then
I.times.I x: R{circumflex over ( )}.sub.3.fwdarw.R{circumflex over
( )}.sub.2.times.[0,1] folds three-space into the unit slab.
[0170] Suppose .gamma.: R.fwdarw.R{circumflex over ( )}.sub.2 is a
sine curve parameterized by arc length, and so .gamma..times.I:
R{circumflex over ( )}.sub.2.fwdarw.R{circumflex over ( )}.sub.3
embeds a corrugation surface. Choose a rotation .rho.: R{circumflex
over ( )}.sub.3.fwdarw.R{circumflex over ( )}.sub.3 so that
.rho..multidot.(.gamma..times.I)(R{circumflex over ( )}.sub.2) is
as in FIG. 15a. Lastly compose with I.times.I.times. so that
(I.times.I.times.).multidot.p.multidot.(.gamma..times.I)(R.sub.2)
is as in FIG. 15c. A priori the surface has zero-curvature; the
double periodicity is verified easily using the periodicity of and
.gamma.. In FIG. 15b, to illustrate the idea has been replaced by
the absolute value function .parallel.. The choice of , .rho. and
.gamma. in the construction are not constrained--other choices
yield other DPFs. The only precaution is that the composition have
the correct periodicity and be injective R{circumflex over (
)}.sub.2.fwdarw.R{circumflex over ( )}.sub.3.
[0171] This example can be generated by the other methods, as seen
in FIG. 4C1, where .gamma.(R) is the RCS. If .gamma.(R) is replaced
with a piecewise linear wave, such as in FIG. 28, W1-W8, and p is
any non-orthogonal rotation of R{circumflex over ( )}.sub.3
parallel to the x-axis, the result will be in the simple class. The
process could also be reiterated again so that
(I.times.I.times.).multidot..rho..sub.2.multidot.(I.times.I.times.).multid-
ot..rho..sub.1.multidot.(.gamma..times.I)(R.sub.2)
[0172] will be a folding tessellation if the rotations .rho..sub.1
and .rho..sub.2 are aligned so that the result is embedded and
periodic. The key ingredient above is that the easy one-dimensional
local isometry was extended to a three dimensional local isometry
I.times.I.times.. Any other one-dimensional local isometries can
also be substituted into the formula.
[0173] To illustrate another composition of local isometries,
select o: [0,1].times.R.fwdarw.R{circumflex over ( )}.sub.2 to be a
strip map (with the switch of coordinates). Then I.times..sigma.:
R.times.[0,1].times.R.f- wdarw.R{circumflex over ( )}.sub.3 is a
local isometry. Select .gamma.: R.fwdarw.R.times.[0,1] to be a PL
local isometry (serving as the RCS) that projects invectively.
Then
(I.times..sigma.).multidot.(.gamma..times.I)(R{circumflex over (
)}.sub.2)
[0174] is a DPF, also producible by the Strip Map Method with CSM
.sigma. (usual coordinates) and RCS .gamma.. But here if .sigma.2
is another stripmap and if .rho..sub.1 and .rho..sub.2 are
rotations chosen within bounds then
(I.times..sigma..sub.2).multidot..rho..sub.2.multidot.(I.times..sigma.).mu-
ltidot..rho..sub.1.multidot.(.gamma..times.I)(R{circumflex over (
)}.sub.2)
[0175] will be a DPF.
[0176] The inventor wrote a computer program to compute these
compositions of local isometries, and return a DPF map R{circumflex
over ( )}.sub.2.fwdarw.R{circumflex over ( )}.sub.3. By adjusting
rotation angle the resulting surface may be given either a periodic
or quasi-periodic structure, essentially due to the harmonic
patterns between the composing maps.
[0177] 8.2 Computations
[0178] Computer programs composing these simple local isometries
can be written by persons of ordinary skill in the art. To study
the results graphically however, computer mesh-point distances may
be larger than the detail of F: R{circumflex over (
)}.sub.2.fwdarw.R{circumflex over ( )}.sub.3, totalling missing the
structure of some of facets of the DPF. This is a consequence of
the likelihood for composing maps to interfere quasi-periodically,
and generate some very small facet regions.
[0179] The significance of our method is how we generate the
vertices of the DPF, to maintain full detail. Making vertices
explicit is nearly essential (versus continuous function approach)
since the detail in the facet geometry with experimental
compositions could be quite small (general would have limit of
zero, by quasi-periodicity). To do this the singular sets of each
of the composing functions must be given a data structure. It is
difficult to use a simple array structure to define the vertices of
the singular complexes because the vertex mesh becomes askewed with
each composition, so a cw-complex data structure may be the easiest
to implement. In general for two composing maps one can calculate
the intersections of singular complexes of adjacent functions in
the composition, to determine the cw-complex data structure for the
composition. Calculating the vertices in the intersection requires
searching for which lines and hyper-planes intersect between the
two maps. This is described below, and then a simpler method is
described exploiting the fact that the composition always starts
with R{circumflex over ( )}.sub.2 on the right.
[0180] A first approach is to enter the maps defined by giving the
correspondence between the vertices of the singular sets before and
after applying f.sub.i, with the edge, polygon, and polyhedron
structures given in terms of these vertices and the full map
defined by isometric interpolation as in Section 7. Then for
f.sub.2.multidot.f.sub.1: R{circumflex over (
)}.sub.3.fwdarw.R{circumflex over ( )}.sub.3 one calculates the
intersection I of the singular set in the domain of f.sub.2 and the
singular set in the range of f.sub.1, and use f.sub.1{circumflex
over ( )}(-1)(I) and f.sub.2(I) to construct the singular sets in
the domain and range respectively of f.sub.2.multidot.f.sub.1. This
enables f.sub.2.multidot.f.sub.1 to be represented by giving the
correspondence between the vertices of the singular sets before and
after applying the new function. By repeating the procedure, longer
compositions f.sub.n.multidot. . . . f.sub.1 can also be
represented by the coorespondence between the singular sets of the
domain and range.
[0181] Our prefered method is a variation of above, that leaves the
composing maps in their original representation, and calculate
f.sub.n.multidot. . . . f.sub.2.multidot.f.sub.1: R{circumflex over
( )}.sub.2R{circumflex over ( )}.sub.3 on the singular set working
from the right. Since f.sub.i has a two dimensional domain this
will simplify the calculations considerably. Many of the other
f.sub.i are rotations or translations and they can be left as
matrixes or vector addition. The other three dimensional local
isometries f.sub.i: R{circumflex over (
)}.sub.3.fwdarw.R{circumflex over ( )}.sub.3 will be products of
one or two dimensional local isometries, with a product structure
on their singular sets. As the Method calculates
f.sub.1(R{circumflex over ( )}.sub.2),
f.sub.2.multidot.f.sub.1(R{circumflex over ( )}.sub.2),
f.sub.3.multidot.f.sub.2.multidot.f.sub.1(R{circumflex over (
)}.sub.2), . . . the image is a two-dimensional surface with a
polygonalization. When the polygonalization for f.sub.k-1.multidot.
. . . f.sub.1(R{circumflex over ( )}.sub.2) has been calculated,
and f.sub.k is a translation or rotation, f.sub.k is applied simply
to the coordinates of the vertices in the polygonalization.
Otherwise f.sub.k will be similar to a stripmap and the new
vertices and polygonal structure can be calculated as in FIG. 13
with mild adaptions. In short since the local isometry
F:=f.sub.n.multidot. . . . f.sub.1:R{circumflex over (
)}.sub.2.fwdarw.R{circumflex over ( )}.sub.3
[0182] starts with a two-dimensional f.sub.1 and continues to use
global isometries or isometrics with simple product structure, the
calculation for the cw-structure on the singular set may be reduced
nearly to the two-dimensional case.
[0183] 9 Inter-Relationships, Similarities, and Blends
[0184] In certain cases it may be advantageous to inter-blend two
or more of our methods or methodologies. For example, potentially
the user could want to use the row data from the Wave-Fold Method
(RED) because of its definitive description of the row of edges on
the folded DPF for perhaps bonding on another laminate, and the
user may want to use the column data from the Strip-Map Method
(CSM) for its versatility or to control the YZ-plane silhouette of
the DPF, and the user may already have the geometrical calculations
of the Two Cross-Section Method (intersecting offset lines) on
their existing software.
[0185] The similarities among the features of these five
methodologies in many cases enable one to translate from the
features of one procedure or method to the features of another.
These features include the type of input data, the geometrical
procedure, the computational procedure, and the range of DPFs
produced. For instance as explained shortly their types of row data
are related by trigonometric formulas, as are the CCS and CET, and
each CCS can be converted to a CSM. Furthermore their geometrical
procedures are very compatible, and allow for blending of the
various methods. This enables easy hybrid of methods, customized to
meet the requirements of a specific application.
[0186] Two geometrical procedures were given for the Two
Cross-Section Method, namely the splicing together of facet rows
and an offset procedure for making the set of column cross
sections. The Wave-Fold Method gave a recursive procedure for
scaling the amplitude of successive copies of RED to get parallel
column edges and a non-recursive procedure that used trigonometric
relationships involving the half included angle of the vertices of
CCS. For the main class of DPFs, the geometrical procedures for
these two procedures are closely related in that intersecting
points of the offset lines will always lie in the plane of the
copies of RED and are equivalently determined by finding the
scaling factor for RED. Moreover the location of these points may
also be understood and calculated by a reflection phenomenon, which
is roughly the approach of the Strip-Map Method. FIG. 16 compares
the geometrical procedure of these three methodologies. The dotted
line represents a column of edges on the folded DPF (or any
calculated column cross section). The similarities between the
geometry of the methods can be deuced from the conguence of the
figures' resulting dotted lines. Note in each case the x-coordinate
of the row data is unchanged in the generated DPF, and thus it is
sufficient that the figure only show the procedure in some
YZ-plane.
[0187] Specific calculations, whether using intersecting offset
lines, intersecting offset lines with reflection lines, applying
reflections to offset lines, calculating scaling coefficients in
normal direction to CCS vertex bisector, and others are also
interchangeable after applying basic plane geometry identities.
[0188] As for the Wave-Tessellation Method, it may be seen as the
limiting case for the Wave-Fold Method. Also after calculating the
vertices of a DPF in the main class by any method, one of our
computer programs will determine the original tessellation. To do
this it calculates the edge lengths on the DPF and the facet
included angles, and then reconstructs the tessellation on the
plane. Alternatively one may convert directly to RET and CET with a
procedure similar to reversing the procecedures of Section 16.
[0189] For the main class the CSM data is very similar to the CCS
data. To convert a CSM data to CCS data, choose a z.sub.0 in the
strip, and apply the strip map to R.times.z.sub.0. The result will
be a CCS capable of generating the same DPF. To convert CCS to CSM
data, apply the Two-Cross Section Method using CCS and a RCS
consisting of a single edge [(0,z.sub.1),(0,z.sub.2)] where z.sub.1
and z.sub.2 are the minimum and maximum values for a RCS that the
CCS can handle. The result will be a folded strip, that before
folding was R.times.[z.sub.1,z.sub.2]. Besides offering different
conveniences on the main class, the difference between CSM and CCS
lies in the way they generalize outside the main class.
[0190] The Composition of Local Isometries Method generalizes the
Strip Map Method. A strip map is a two dimensional local isometry,
and its product with the identity on R gives a three dimensional
local isometry used for the compositions. Additionally the
Wave-Fold Method defines a plane normal to the bisector of CCS,
that is used in calculations as if it were a reflecting plane,
where the process of reflecting is also a local isometry. However
the Composition of Local Isometries Method allows for reiterations
and other procedures.
[0191] To facilitate the conversion between the geometric
procedures performed by the algorithms and methods, we offer the
following explanation. If a line in the plane is reflected
(envisioned as a beam of light) at some acute angle between two
parallel mirrors in the plane, it will produce a zigzag shape (FIG.
18a). More complicated repetitive reflecting patterns for a line
can be produced by applying a series of mirrors (FIG. 18b).
Effectively, the CCS serves as the reflection pattern. Three of
these reflection patterns are labeled "1,2,3" in FIG. 4. Reflection
preserves arc length, so the resulting map induced by the
reflections is a local isometry of the line. It is also possible to
put a band around a reflection pattern for a line, and get a
strip-map.
[0192] Likewise to reflecting a line, a plane can be reflected by
adapting the mirror arrangement to three-space. For example FIG.
19a is obtained by extending the reflection scheme of FIG. 18a.
This will produce a conventional folding pattern for the plane. The
resulting map induced by the reflections is a local isometry of the
plane but the image has only one direction of periodicity and so is
not a DPF. Instead of sending a plane to reflect between repetitive
mirror arrangements, one may apply the mirrors to a
corrugation-type surface (FIG. 19b).
[0193] The corrugation type surfaces can be specified by their
cross section. Three of these cross-sections are labeled "A,B,C" in
FIG. 4. The body of FIG. 4 can be interpreted as showing the result
of applying the reflection schemes 1, 2, and 3 (labeled `column
cross sections` in the figure) to the corrugations having
cross-sections A, B, and C.
[0194] To further facilitate and disclose how to mix and blend
features of our various procedures and methods, namely their row
input data type, their column input data type, their geometrical
procedures, and their calculation procedures, FIG. 17 is included
showing the trigonometric relationships of RCS, RET, RED, CCS. With
this information the user can customize our procedures and methods
by mixing and blending their features.
[0195] FIG. 17 shows the relationships between the entry data for
generating a single row-edge e on the DPF. The boldface segments
under the headings RCS, RET and RED are the edge of the
corresponding row entry data used to generate e. The additional
triangles show the rise and run of the segments on the row data
plane. Note the x,y, and z axis are labeled to agree with the
convention for entry data established earlier, and become re-scaled
and rotated differently under the various headings of the figure.
The CCS vertex used to generate the row-edge e is shown in boldface
on the bottom piece of the figure. The perpendicular bisector and
the normal to the bisector are shown as dashed lines. The one-half
included angle .gamma. is marked, and again in a congruent
location. The right triangle formed using the dotted line S is
reproduced for clarity on the lower right-hand corner.
[0196] The quantities C,D,E,S,T,U,.sigma.,.tau.,.upsilon. also do
have geometrical significance. The angles .sigma.,.tau.,.upsilon.
are seen on the entry data, as the angle of inclination of the row.
The distance between the endpoints of e is E. The distance between
the column segments, extended to parallel infinite lines,
containing the endpoints of e is C. The distance between planes
normal to the x-axis of the DPF and containing the endpoints of e
is D. The lengths S,T,U appearing in under the headings RCS, RET,
and RED are the (usually signed) rise in the entry data as shown in
the right triangles. They appear again in the CCS right triangle in
the lower portion of the figure.
[0197] With this the data types become quickly
inter-formulated.
[0198] Another similarity of these five methodologies and their
derivatives, is that they all generate DPFs by combining an
infinite selection of row data with an infinite selection of column
data. The fact that our Main Class has this array structure that
indexes by rows and columns is novel and versatile.
[0199] 10 Extension to the Main Class
[0200] The procedures and methodology discussed so far has focused
on DPFs in the simple class. DPFs in the simple class have a
tessellation with four-sided regions meeting four to a vertex, have
edges that are straight line segments, and have an overall
slab-like shape in the sense that it that fits between two parallel
planes. The main class is very similar to the simple class, but may
have triangular regions, curved edges, and/or an overall shape that
is cylindrical for instance. DPFs in the main class can be produced
by all of the procedures and methodologies.
[0201] It is easiest to extend from the simple class to include
triangular regions in terms of the Wave-Tessellation Method.
Consider two adjacent "waves" in the tessellation, with differing
amplitudes. (The easiest case to see is with amplitudes of opposite
sign.) By lessoning the spacing between these waves until they
coincide at some points without crossing, the DPF will gradually
change with certain facets getting smaller until they disappear
entirely. Facets in the same row adjacent to the disappearing
facets will become triangles. Some vertices will increase in
degree.
[0202] This can be accomplished in corresponding form by each of
the methods. In the Two Cross-Section Method, if the RCS is tall
enough the offset will be large enough (see FIG. 5) so that one of
the offset segments calculated will have zero length. In the
Strip-Map Method, if the strip has triangles or internal vertices,
it is possible for the DPF to have triangles provided the RCS is
positioned inside the strip to meet the vertices of exceptional
degree. (Exceptional degree here means a horizontal open half-plane
through a vertex contains two edges of the vertex.) The other
methods produce triangular regions similarly.
[0203] To generate DPFs with curved facets by any method, simply
use a non-linear row input data. For instance in FIG. 4C, a sine
wave y is used for RCS with three different examples of CCS. FIG.
15a shows the a sine wave corugation surface R.times..gamma.=S with
the application of the local isometry in FIG. 15c. FIG. 4C1 and
FIG. 15c are essentially the same DPF, which could also be produced
by applying the top strip map in FIG. 11 to this S. Other curves
can be used for the row data RCS,RED,and RET, as well as pieced
together linear and curved data.
[0204] For cylindrical overall shapes of the DPF, select a CCS on
the YZ-plane with an overall cylindrical shape. The CCS may be a
hexagon, an octagon, and other non-intersecting star shapes for
example. This corresponds to a strip map of a finite strip that
rejoins at the end. Arch-like DPFs can be consturcted using
arch-like column data.
[0205] DPFs in the main class can be described by all of the
designing methods of Part I. There are additional extensions to the
methods outside the main class. Some of the methods have extensions
that do not easily apply to the others. Because of the
inter-relationships between the various methods described in
Section 9, in some cases or for some input data it will be possible
to translate to another of our methods.
[0206] There are many traditional forming technologies that can be
combined with our technology, such as perforating or re-enforcing
the sheet before or after forming, either generally across the
sheet or in coordination with the facet positions, splicing DPFs in
design or material with other sheet products, and using a DPF as
architectural base providing the substantial sheet placement, with
deforming processes super-imposed.
[0207] Part II Selected Structures and Features of DPF
Technology
[0208] 11 DPF Data and Pattern Type Parameterizations
[0209] This section demonstrates the use of our DPF technology to
study a DPF pattern type through parameterization. This is key
value when optimizing a given pattern type for a given application.
Our method will be illustrated first for the DPF pattern type in
FIG. 41A, called the chevron pattern in the literature. The task is
to represent the chevron pattern by convenient parameters, and
study the effect of parameter variations. In this example the
pattern will be used as a core material between two laminated
faces. Since the rows of edges in the folded DPF are easily
recognized as the ridges and valleys, and they describe the contact
area to be glued to the faces, in this example we optionaly chose
RED for our column data type. The column data type CCS is selected
because it imidiately gives both the pitch (the spacing between
successive ridges) and the thickness of the chevron pattern. RED
and CCS are shown in FIG. 25 upper and lower portions of the
figure, respectively. The first segment of both waves, shown in
boldface in the figure, has been positioned with its initial end on
(0,0). This way the parameters of the other endpoint of the segment
entirely determine the segment and thus through symmetry the the
entire wave. The coordinates of the other vertices are shown. Thus
RED is entirely determined by (a,b), and the CCS is entirely
determined by (B,C), and the chevron pattern type has been reduced
to independent vertex parameters (a,b,B,C).
[0210] To generate the DPF vertices, one may subsitute values into
(a,b,B,C), calculate RED and CCS (or their units), and then use the
Wave-Fold Method to generate the DPF with those parameters.
Alternatively, one may run the un-evaluated parameters through the
method to determine a variable description of the DPF vertices. To
do the latter, since the plane normal to the bisectors of the
vertices of CCS are all parallel to the XY plane, the scaling
factors for all the copies of RED are all 1. With this the
Wave-Fold Method produces a parameterized description of the
vertices of the folded DPF, as depicted in FIG. 26. The (x,y,z)
position of the vertex in the mth row and the nth column of edges
is given in the table in FIG. 27. Note the rows run in the
x-direction and the columns run in th y-direction, and the data was
as in FIG. 25 so that the zeroth row and zeroth column vertex is at
(0,0,0). FIG. 32 shows the chevron pattern under various values of
(a,b,B,C) where
[0211] a=x-extension of a row edge
[0212] b=y-extension of a row edge
[0213] B=1/2 pitch
[0214] C=thickness
[0215] interpreted directly from the definitions of RED and
CCS.
[0216] Of course parameters other than (a,bB,C) for the chevron
family of DPFs also can be given by using RET or RCS instead of
RED, by using CET or CSM instead of CCS, and/or applying obvious
identities such as using polar coordinates to express the entry
data.
[0217] This procedure for expressing an entire family of folding
tessellations in terms of a list of parameters applies to all DPF
pattern types by following steps similarly to the example above.
One choses the form of the row and column data most meaningful to
the application, expresses the row and column data as parameters,
and uses the methodology of this disclosure to generate DPFs of the
selected pattern type. FIGS. 28,29,30 show many waves that can be
easily reduced to independent parameters. Waves 1-9 and 13-19 21,
22 may be used for CCS, Waves 1-8, 10-12 and 20-26 are intedned for
RCS, or RED, and 1-3, 6-8, 10,22,23,25,26 for RET. Section 14 says
many ways the waves are used to generate DPFs.
[0218] 12 Mounting Plate Application and Design
[0219] The physical DPFs have many applications, including
compartmentalization of space and structural core materials. In
these applications and others the DPF often must be attached to
external mount points or laminates. It is sometimes useful to
design the DPF with convinient locations for contacting the other
object. This may provide areas for gluing, welding, riveting, or in
general bonding the two together. These areas may be entire facets
or partial facets of the DPF, and are called mounting plates. For
instance, one application is attaching a DPF truck bed to the truck
frame. The mounting plate has many uses including attaching
brackets or other objects conviniently, and having flush contact
regions for bonding a folded core material to laminated faces. Note
the competing honeycomb core material only bonds on its edges to a
laminated face, as do prior folded core materials such as FIGS. 4A1
and 4C1.
[0220] To incorporate a mounting plate into a DPF parallel to the
XY plane, a horizontal segment ((a,f),(b,f)) can be included in the
row data, with another horizontal ((c,g),(d,g)) in the CCS data.
This will produce a horizontal rectangular facet of width (b-a). In
the case that f=0, the height and length of the facet will be g and
d-c, respectively. For other f the height and length are adjusted
by offset corrections as shown in FIG. 16, where the quantities
depend on the choice of row data (RCS, RED, RET), the included
angle at (c,g) and (d,g) on CCS, and the sign of C(-1){circumflex
over ( )}n, where n is the vertex (c,g) in the CCS vertex list.
These quantities can be determined using the DPF Methodology of
Part I. In FIG. 5, the procedure for determining offset using RCS
row data type. FIG. 4b3 shows an application of this method for
designing horizontal mounting plates to bond to parallel faces. In
the figure the segments generating the mounting plates occurred
periodically in the row and column data, yielding an array of
mounting plates.
[0221] It is also valuable to use sloped segments in the row and
column data to design non-horizontal mounting plates. To design a
mounting plate in a given location one may determine the row-cross
section and the column cross section of the desired plate and
spline the segments into the RCS and CCS data. Mounting plates are
also useful in many multi-layer laminations, that use folding
tessellations exclusively or combine corrugated surface, plane
surfaces, or other laminates.
[0222] The design and use of mounting areas for materials formed by
traditional methods such as stamping and casting is well
established. For folded tessellations and other complex folded
structures the use of mounting plates has similar utility. Not only
is this a new art for folding tessellations, but we present
designing techniques for incorporating mounting plates into any of
the folded structures in DPF technology.
[0223] 13 Tie Areas
[0224] In application DPFs may be needed to meet specific physical
demands, including strength and vibration absorbtion. For these
needs and others it is valuable to incorporate Tie Areas in the
folded tessellation structure. These are facet-to-facet bonding
areas within a folded structure. Viewing a folded tessellation as
an assembledge of individual facets, forming the structure by
folding a sheet is already an extrememly efficient procedure, for
the connectedness of the original sheet enherently provides the
facet-to-facet connection across shared edges. Tie areas are
additional bonding areas within the DPF that provide locations for
gluing, welding, riveting or otherwise bonding the folded
tessellation to itself. The manufacturing procedure for bonding the
tie ares may occur in parallel or after the forming process, or in
certain applications such as producing cylindrical DPFs it may be
done in advance.
[0225] One technique for producing tie areas is to design the
folded tessellation to have edges positioned so their included
angle is 0 degrees, that is the edge is completely folded. In this
way interior portionss of the facets sharing the edge are in flush
contact, with this area being ideal for bonding. The technique can
be applied across row edges and across column edges in DPFs. Tie
areas can also be utilized to bond facets that do not share an edge
of the tessellation. In FIG. 28 waves such as W9 can be used as CCS
to accomplish this.
[0226] To design tie areas to bond adjacent facets sharing a common
row edge, one may use RCS or RED row data with a vertical edge. The
FIG. 10 shows entire columns of facets, generated with a vertical
RCS or RED edge, all successively connected across tie areas. The
figure also shows the column under our Uniform DPF Process. A
detailed description of this type of tied column is given in
Section 16.1.
[0227] The Strip-Map Method may also be used to design tie areas by
using a strip map devised to fail the infectivity condition on its
edges. This can produce tie areas both for facets that share an
edge and for those that are not adjacent on the tessellation.
Previously folded core materials had relied on the connectedness of
the original sheet for their physical properties. Folded
tessellation having tie areas offer additional possibilities of
bonding or fastening the folding tessellation to itself. Advantages
include stronger core materials.
[0228] 14 Specific Patterns
[0229] This methodology can generate countless DPFs with a wide
assortment of applications. FIG. 28,29,30 show many wave types,
which can be readily parameterized as explained in Section 11. For
RCS and RED data W1-W8, W10-W12, W20-W26 are useful, and for CCS
W1-W9, W13-W19, W21, W22 as well as many not drawn. FIG. 4 shows
the array of patterns generated by a few of these waves.
Furthermore there are many variations obtained by changing the
parameters.
[0230] For RCS W4, W5, W12, W20, W21, W24 will produce DPFs with
tie areas. In FIGS. 1D and 4B1 the combination RCS:W4 and CCS:W1 is
used to produce a core material that becomes exceptionally strong
when bonded on the tie areas. A designer may antipate analytically
mixing this core material with the chevron pattern, by trying RCS
waves such as W5, W20, W21. These examples may demonstrate a
tequnique for strengthening the sides of a core panel, with W21
showing different flexural properties from above and below. Using
RCS:W26 and CCS:W4 or CCS:W3 produces an intersetingcushioning
material with many mounting plates on one side and round flutes on
the other, perhaps for easily installing a cushioning barrier.
[0231] For some applications such as packaging materials, a product
may manufactured to a volume reduced state, shipped and stored
until purchased by a consumer, whereupon the consumer expands the
material to its usable state, and puts it to application. For RCS
W1,W6,W8, and CCS W1-W8 the material may manufacure to almost a
solid block, to save volume related expenses. For cylindrical
materials, CCS W14-W19 offer a interesting results. The odd shaped
polygons need to be spiralled slightly. This has spiralling has
value also for even sided CCS, for assembling a long faceted tube
from narrow sheet material. Combining CCS:W119 with many RCS waves
will produce a core material for interfacing between two cyllendar
faces. CCS:W13 produces another structural pattern for arches or
curved panels.
[0232] Part III Forming Process and Material Flow
[0233] 15 Sheet Process Background
[0234] There are many machines and procedures to manufacture the
three-dimensional structures produced by our patterning technology.
Some include casting, cutting, assembling and stamping. A
significant value of DPF structures is that their intrinsic surface
geometry corresponds to the intrinsic surface geometry of a sheet.
This offers the intended possibility of manipulating a sheet into
the DPF geometry without significantly stretching the sheet
throughout the forming process. The main obstacle is because none
of the folds extend clear across the sheet, all of the fold
vertices impose simultaneous constraints forcing the facet motion
of the entire sheet to be interlocked. Additionally, the process
should be reasonable to implement in machine design.
[0235] In origami texts, a process of forming the "chevron" pattern
can be found, that folds, unfolds, back-folds, re-folds, etc a
sheet of paper one move at a time until the whole chevron pattern
is folded. In this procedure no extraneous creases are made, but
many creases are folded and refolded with opposite convexity many
times. Anther option for a forming process include allowing
temporary folds or bends in the material in the interior of the
facet regions that are later removed. In Section 17 another process
is disclosed, of allowing a fold to migrate or roll as it is formed
until it moves into its final position and then becomes clearly
instated. Combining these techniques gives countless potential
processes for forming folded structures.
[0236] In general one process, described by the three-dimensional
flow, can be implemented by many machines. There are many
apparatuses to move and direct a material through a cascading
network of geometry. However the process, the sequencing and
specification of the sheet motions, is essential for assuring the
relative distances measured within the material between any two
points remain constant, to prevent any stretching of the sheet
during the forming of the DPF.
[0237] We have developed three processes for producing DPFs without
significantly stretching the sheet, and built machines that
successfully implement them. To distinguish our procedures from
others we have called ours the Uniform DPF Process, the Novel
Creasing Process and the Two-Phase Process. The first applies to
all DPFs in our main class and to some DPFs outside the main class,
but generally will not apply to folding tessellations designed
outside our methodology. The second process describes a novel
method of producing an individual crease locally. This creasing
process is especially useful when faced with the task of
efficiently folding complicated networks of folds. The third
process solves the tessellation folding problem from a very
practical production point of view, and applies to many folded
sheet structures inside and outside of our methodology.
[0238] 16 Uniformi DPF Process
[0239] 16.1 General Information
[0240] This section offers a process that solves the `linkage
constraints`. These constraints imply the material is not stretched
during the forming process, that no temporary creases, bends or
folds occur in the middle of facets during the forming process, and
that the folds creases do not migrate. An advantage of this type of
forming process is that other forming processes that do not satisfy
the linkage constraint essentially change the crease tessellation
during the forming process, and in many cases it is desirable to
have no temporary folding. The only activity that occurs during the
forming processes under the linkage constraints is the folding
along the edges of the tessellation and the corresponding rigid
motion of the facets. In our solution, the Uniform DPF Process, the
fold angles along the creases of a DPF are increased gradually, and
simultaneously across the entire tessellation.
[0241] Most of the prior state of the art knowledge on folding
tessellations has been done on small paper models. In these models
the individual facets do not always remain planar, and
generalizations about the folding process are easily made
erroneously, and so will not apply to larger tessellations or the
tolerances of a machine. One mistake is that if a tessellation has
a three-dimensional folded position, to expect it to have a gradual
folding process that satisfies the linkage constraints. In FIG. 20,
a four-sided polygon is drawn with 16 angles a through p labeled.
The polygon may be a square, a rectangle, a parallelogram, a
trapezoid, or a general quadrilateral. The angles drawn are not to
scale, and the figure is intended only to represent a candidate for
a degree-four region, with degree-four vertices, in a arbitrary
tessellation. Nearly all (the exceptional set has co-dimension at
least 2 and may be closer to 10) of these components will not fold
gradually. Unless all of the angles are selected in coordination,
similar failure will also occur for triangle, pentagon, and other
polygon components with degree four vertices. For instance if a
linkage with hinges was constructed to imitate one of these
component, forcing the linkage to fold would cause it to bind and
bend the leaves of the hinge, or otherwise distort the device. For
a generic component correcting the failure would involve changing
several of the angles in the figure whose calculation could involve
simultaneous quadratic equations involving over 20 variables.
Furthermore for a tessellation to fold gradually, not only must
every component fold gradually, but all of the components must fold
gradually with interlocking equations. The DPFs in the Simple Class
have all of their polygons represented by the figure, and will fold
gradually under the linkage constraints because of the unusual
geometry of our designing algorithms.
[0242] DPFs described by row-column methodologies in this
disclosure have a distinct advantage over folding tessellations
generated by means outside our technology. It has been stated that
it is difficult to design folding tessellations without a
methodology, for the no-stretch zero-curvature condition imposes a
very complicated set of simultaneous conditions on the vertices and
edge lengths of a proposed structure. Once designed, as explained
above, determining a procedure to fold a tessellation can be even
more difficult, even if the description of the desired
three-dimensional form is precisely known. In fact most often the
linkage constraints have no solution. For DPFs in the Main Class
defined by our methods, not only does our Uniform DPF Process
satisfy the linkage constraints, but we have numerical procedures
that give the Uniform DPF Process. The technology enables one to
model the material flow inside a machine and to calculate the
position of the various apparatus or computer controlled mechanisms
that guide the material.
[0243] For DPFs in our Main Class the Uniform DPF Process can be
specified in terms of any of our designing methods. With our
Uniform DPF Process is associated an additional parameter t, called
the flex parameter. The input row and column data can be adapted to
yield the correct surface at the intermediary stage t of forming.
This is only possible because the intermediary surfaces formedfrom
DPFs in the main class by the Uniform DPF Process are also DPFs in
the main class. This is another unique utility of using our
methodology.
[0244] Each of the DPFs in the main class may start as a
tessellation in the plane and may be folded, by the Uniform DPF
Process, to increasing extent until a there is a collision of
facets within the structure. Often the collision occurs between two
adjacent facets when there common edge is folded to 180 degrees (0
degree included angle). Any earlier case of facet collision would
happen between facets in the same column, and can be ruled out by
checking the embedding condition in the Strip-Map Method. When the
collision occurs between two adjacent facets, there will in fact be
an entire column of facets with their bridging edges folded to 180
degrees. Moreover the entire column will lie in a single vertical
plane, and the segment of RCS generating the column will be
vertical. Moreover during the Uniform DPF Process the angles of
inclination or declination of the segments of RCS start at 0
degrees and change monotonically until one of them reaches the
vertical position (assuming as before the strip map is injective in
the first coordinate). For this reason we have elected to choose
for our flex parameter t the parameterization that runs from 0 to
90 degrees and represents the maximum of the absolute values of the
angles of inclination of the segments of RCS. In FIG. 17 this would
be the maximum of .vertline..sigma..vertline. over all edges of
RCS. Of course t could be parameterized or re-parameterized in many
ways, for example simply composing this parameterization on the
left or right with any smooth monotone function, but for
definiteness this t is the parameterization used here.
[0245] As only the tessellation data RET and CET remain unchanged
during the folding process, while RCS, RED, CCS, and CSM change
trigonometrically, we will first show our method for specifying the
Uniform DPF Process in terms of RET, CET, and t. This is valuable
also because in Section 4 a procedure was given for producing
tessellations that would produce the Main Class of DPFs, but no
method (other then experimentally working the structure) was given
for calculating the three-dimensional form.
[0246] 16.2 Uniform DPF Process from Wave-Tessellation Data
[0247] To utilize the advantage of the zero-curvature structure of
DPFs, the forming process also should not induce significant
in-plane stretching of the material even temporarily. The forming
process may fold, bend, and perhaps unfold, unbend the material,
but should not produce in-plane distortions beyond the requirements
locally in the folds related to the material thickness and fold
radius. Of course if the DPF technology is being used as an
architectural base to design hybrids in between traditional forming
technology and folding technology, the no-stretch condition during
the forming process may be relaxed proportionally to the hybrid. In
either case it is desirable to have a mathematical description of a
folding process that precisely maintains the no-stretching
condition, whether applying it to manufacture a structure produced
strictly by folding or using it for an architectural base for a
hybrid process.
[0248] Several methods for calculating the position of a
tessellation in three space during the Uniform DPF Process are
described in the subsections that follow. Other defining data for a
DPF (RCS,RED,CCS, CSM and the three-dimensional image) all change
continuously throughout the Uniform DPF Process, but the
tessellation and tessellation data remain constant. Thus the
Wave-Tessellation Method provides information that is valuable
throughout the Uniform DPF Process, while the Two Cross Section
Method, the Wave-Fold Method, the Strip Map Method, and the
Composition of Local Isometries Method only give a static DPF,
unless they are additionally parameterized and supplied with data
indexed by a time or the flex parameter.
[0249] This first method converts RET and CET to RCS (or RED) and
CCS in terms of the flex parameter t and then applies the Two Cross
Section Method (respectively the Wave-Fold Method) to generate the
three-dimensional folded form. With the background knowledge of the
procedures and methodologies in this disclosure, the most direct
approach to generate the folded structure for RET and CET at time t
is to convert the data to RCS/RED and CCS and apply the appropriate
method. As in FIG. 17, let E.sub.i and .tau..sub.i represent the
length and angle of inclination of the ith edge of RET. Likewise
for the other variables in the figure under RCS, RET, and RED, the
subscript will correspond to the depicted in the ith column in the
row. The subscript m is first determined so that the mth segment of
RET has maximal absolute slope, that is
.vertline..tau.m.vertline.=max(.vertline..tau..sub.i.vertline.),
where i runs over the edges of RET. We will also use Cm, Tm, and
Em. For CET the subscript j will denote the jth vertex and
augmented pair (yj,aj) as in Section 4.4, where yj is the spacing
on the y-axis between the the (j-1)th and jth row wave of the
tessellation, and aj is the amplitude factor for the jth row wave.
Once folded to parameter t, the angle yj will denote one half the
included angle of the jth vertex of CCS similarly to depicted in
FIG. 17.
[0250] At t=0 RCS is collinear and horizontal with its ith segment
of length Ci, determined as the x-displacement of the ith segment
of RET. At time t the x displacements and column cross section
elevation of the ith segment of RET are Di and Si respectively.
Define Sgn:R.fwdarw.R to be the function that returns -1,0, or 1
according to the sign of the variable. Solve for
Sm=sin(t).multidot.Cm.multidot.Sgn(Tm)
Dm=cos(t).multidot.Cm
[0251] Put k=Sm/Tm. Then at time t for the ith segment of RET:
Si=kTi
Di={square root}(Ci{circumflex over ( )}2-Si{circumflex over (
)}2)
[0252] Then RCS is constructed recursively by
(x0,x0)=(0,0)
(xi,zi)=(xi-1+Di,zi-1+Si)
[0253] The method for producing RED is very similar. Put k={square
root}(Em{circumflex over ( )}2-Dm{circumflex over ( )}2)/Tm=Um/TM.
Then the x and z displacement of the ith segment of RED are Di and
Ui respectively:
Di={square root}(Ci{circumflex over ( )}2-Si{circumflex over (
)}2)
Ui=kTi
[0254] The row RED is then constructed recursively by
(x0,z0)=(0,0)
(xi,zi)=(xi-1+Di,zi-1+Ui)
[0255] The method for calculating CCS at flex parameter t is as
follows. Recall CET was expressed as pairs (yj, aj). Put
k=tan(.tau.0)/(a0 sin(t)). Then
Length of jth segment of CCS=yj
.gamma.j=arctan(kaj)
[0256] To assemble CCS an inductive calculation may also be
performed using the lengths yj And half included angles .gamma.j in
the YZ plane after choosing the position of the first edge.
[0257] Alternatively the trigonometric relationships within FIG. 17
can be applied in other orders around the triangles of the figure
to arrive at similar calculation for RCS, RED, or CCS. After
calculating either RCS or RED and CCS, the Two Cross Section Method
or Wave-Fold Method or others generally described in this document
may be applied to calculate the DPF folded to stage t.
[0258] 16.3 Uniform DPF Process with Two Cross-Section Method
[0259] This section completes the steps for taking a RCS and CCS
and the resulting DPF, and according to a new flex parameter,
changing them to produce the correct new RCS and CCS and DPF.
[0260] By using our Two-Cross Section Method, we have described the
process of using our RCS and CCS data to design a DPF. Next we
describe our technique for describing the Uniform DPF forming
process in terms of RCS and CCS. Based on the original RCS and CCS
data, this technique will produce the new RCS and CCS data
corresponding to any flex parameter 0<t<=90. From this the
Two Cross-Section Method may be applied to generate the DPF at
state t. Schematically,
t.fwdarw.(RCSt,CCSt).fwdarw.DPFt
[0261] From the original RCS ((x0,z0), . . . (xn,zn)) one readily
converts back and forth between the incremental form ((x0,z0),
(D1,S1), (D2,S2)) . . . (Dn,Sn)) where D and S are as in FIG. 17
and
xk-xk-1=Dk
zk-zk-1=Sk
[0262] One could alternatively use incremental polar coordinates
(ck,.sigma.k) as in the figure. In either case for simplicity
assume z0=0.
[0263] Select a maximally sloped edge m to give the current flex
parameter,
.sigma.m=max arctan(Sk/Dk)
[0264] Then, for any flex parameter value t, 0<t<=90, compute
the new incremental RCS at stage t in incremental rectangular
coordinates by
Sk'=Sk*sin(t)/sin(.sigma.m)
Dk'={square root}(Sk{circumflex over ( )}2+Dk{circumflex over (
)}2-Sk'{circumflex over ( )}2)
[0265] Or in polar coordinates by
Ck'=Ck
.sigma.k'=arcsin (sin(t)/sin(.sigma.m)*sin(.sigma.k))
[0266] In either procedure the quantity bt=sin(t)/sin(.sigma.m)
does not depend on k and could be substituted in advance into the
calculations. Once RCS is determined it is then converted back from
incremental coordinates to the new ((x0',z0'), . . . (xn',zn')).
The procedure for adapting the original CCS to the new CCS at stage
t is similar. Convert CCS ((y0,z0), . . . (.gamma.N,zN)) to
incremental polar coordinates (Lk,.theta.k) so
Lk*sin(.theta.k)=zk+1-zk
Lk*cos(.theta.k)=.gamma.k+1-yk
[0267] Then
.gamma.k=(180+.theta.k-.theta.k-1)/2
Lk=Lk
.gamma.k'=tan(.gamma.k)*sin(.sigma.m)/sin(t)
[0268] Then with .theta.k'=2.gamma.k+.theta.k-1-180, one converts
CCS from its incremental polar form to ((y0',z0'), . . . (yN'zN')),
using the relationship expressed above. This procedure for
calculating RCS and CCS at flex parameter t can trivially be varied
using the relationships in FIG. 17. Another very similar technique
that we have explained converts RCS and CCS into RET and CET as
explained in Section 9, applies the Uniform DPF Process with the
Wave-Tessellation Data (16.2) to get the DPF at stage t. RCS and
CCS can then be read off the DPF if needed for application.
[0269] 16.4 Uniform DPF Process with Other Methods
[0270] The Wave-Fold Method and Strip-Map Method can also be used
to study the Uniform DPF Process. The procedure is similar to
Section 16.3. The row and column data are entered in advance. The
flex parameter t is entered. The row and column data are converted
to correspond to the DPF at stage t. The DPF at stage t is then
generated from the row and column data. There are many other
alternatives due to the relationships in the triangles of FIG. 17,
including converting the data to apply to Section 16.2 or 16.3.
[0271] 17 Novel Crease Forming Process
[0272] Surprisingly, there are still new methods for forming
individual creases in sheet material. Our novel crease forming
process is useful in situations where the material or forming
apparatus pose geometrical constraints on the material flow. In
particular for folding complicated networks of folds, our new
method for forming individual creases is of great value in that it
allows for unusual combinations of folds to occur
simultaneously.
[0273] 17.1 Traditional Fold Formation
[0274] Naturally, to form a crease one may etch, or mark the crease
location and then carefully fold it. In brakes and other folding
machines one positions the sheet properly and then folds it on the
creasing axis of the machine. For continuous process the sheet
passes through rollers that instate the crease at the desired
location. In each case the folding operation is centered near the
location of the final crease. FIG. 21 schematically depicts three
variations showing the location of the fold as it is formed.
Whether the material is bent softly and then the radius decreased
until the crease is formed or whether the radius is small and the
fold angle increased, the folding is centered near its final
position. In the figure each drawing represents a cross-section of
the sheet material as the crease is formed. In the first series the
curves are polynomial, the second series the segments are filleted
with progressively smaller radii, and the third series is simply an
angle becoming more acute. Of course many other curve profiles
centered at the final crease location are used by traditional
folding processes. The figure both represents a cross-section of
the sheet drawn through and edge crease location and through a
vertex crease location.
[0275] 17.2 Fold Migration
[0276] Migrating folds are almost common. In the home, a wrinkled
piece of aluminum foil may be smoothed by dragging the sheet across
the edge of a sharp countertop. The countertop edge produces a
crease that travels across the sheet. In the garage, on a belt
sander the curved portion of the belt is constantly traveling
across the belt surface. Viewed relative to the belt, the curved
portion rolls across the material. The two examples are similar,
although the countertop edge produces a very small fold radius in
comparison to the radius of the drum of the sander.
[0277] 17.3 Novel Crease Forming Process
[0278] Our process, called the Novel Crease forming Process,
applies both to edge creases and vertex creases. In common
situations it first appears cumbersome, or even absurd, and this
non-obviousness had previously inhibited its invention. However
this procedure has great utility when forming creases in
constrained enviroments, or under unusual circumstances. Our
process combines the two processes above. By usual standards our
process starts by `mis-folding` the sheet and then rolling the bend
or crease into final location. The material is softly curved or
temporarily creased in a position not directly centered on its
final location. The curve or crease is then caused to migrate until
it reaches its desired final location. While it is migrating,
typically the fold radius and or included fold angle are tightened
in preparation of instating the crease at its final location. In
some cases the migrating crease will leave a mark in the material,
but this can be removed by secondary operation later if
necessary.
[0279] FIG. 22 shows two variations on our Novel Creasing Process.
In the first series a soft fold migrates and tightens until it
reaches the final location on the sheet, and the second series
shows a sharp fold migrating and tightening. In the first five
cross sections of the second series both the fold location and the
fold angle change; in the last cross section only the fold angle
was changed.
[0280] One advantage of this for manufacturing networks of folds is
that it allows for the sheet geometry to be `roughed out` first
under much more lax constraints and with more diverse geometrical
tolerancess. In particular, the migrating crease locations allow
for the tessellation to vary during the forming process. For
continuous processes, our novel creasing process may have the
creases migrating into the oncoming sheet material as they are
formed, and greatly reduces the complexity required of the forming
machine.
[0281] An application and advantage of our Novel Creasing Process
is shown in FIG. 23. In the geometrical environment shown,
traditional folding process can produce creases only near the ends
of the material, while our process has much greater
flexability.
[0282] 18 Two-phase Process
[0283] 18.1 Utility
[0284] Our Two-Phase Process solves two problems. Forming fold
tessellations without significantly stretching the material during
forming is challenging. The difficulty of working the material
locally on the folds, while the overall size of the material
contracts in both the x and y directions simultaneously, posses
obvious tooling complication and expense. Moreover for continuous
processes, since one end of the sheet is folded and the other end
in not, some other method not obligated to satisfy the linkage
constraints should be employed.
[0285] 18.2 Continuous Machine Implementation
[0286] In FIG. 24, a schematic of our continuous machine is shown
with the material initially pre-gathered into corrugation at A),
the DPF forming rollers at B), and the finished DPF at C). The axis
of the roller is parallel to the y-axis of the DPF, and the flutes
and the direction of the material flow are parallel to the x-axis.
In this machine the material is taken from a roll and first
pre-gathered into a sine-wave corrugation. The contraction ratio,
that is the ratio of the width (the y distance) of the material
projected onto the AYplane to the width of the material if measured
when unfolded is the same for the material at A) and at C). This
notion of pre-gathering the material to have the same contraction
ratio as the final DPF is essential to prevent the material from
having to contract inside the roller in the direction of the roller
axis. For wide rollers this would require the material to slide in
the y-direction over the teeth while inside the rollers, which is
almost impossible. But contraction in the x direction within the
rollers occurs naturally because the forming region on the material
is advanced in the x direction, and the teeth revolve to engage in
the x-direction. The projected image of the material more rapidly
enters into the rollers than it exits.
[0287] The strategy of this material flow process was to simplify
the simultaneous two-directional contraction of the Uniform DPF
Process into independent x and y contractions. In the first phase,
the y-pre-gathering into a corrugation type material is easy to
perform with a series of guides or rollers; and in the second
phase, the remaining x-contraction and fold crease formation is
performed easily by advancing the material through a simple roller
mechanism. The machine has been tested successfully on paper,
copper, and aluminum. As the DPF is produced without stretching the
material, this is also a novel use of rollers tofold sheet
material.
[0288] In the descriptions above one may also use our Continuous
Machine Implementation with the roles of x and y interchanged
[0289] 18.3 Batch Machine and Process
[0290] A batch process may proceed as the continuous process above
using shorter rectangular sheets. In some cases for rectangular
sheets it may be preferable to advance the rectangles along one
axis first and then along the other axis second. In this way both
phases may induce the contraction in the direction the material is
advancing. This contrasts the continuous process above, where since
the material is continuously moving in the x-direction, the initial
y-contraction occured orthogonal to the material flow. For machine
implementation, a rectangular sheet may be passed first through a
mechansism in they direction, and then through another mechansim in
the x direction, as shown in FIG. 31. The first pass imparts a
corrugation to the sheet with flutes running parallel to the
x-axis. The first mechansim in simple cases could be a pair of
corrugated rollers. Since the first phase only requires the overall
size of the sheet (the projected image onto the XY plane) to
contract in the y direction, and the material is advancing in the y
direction, the contraction can take place in the forming region in
the corrugated rollers. The produced corrugation is desired,
however, to have y-contraction ratio the same as the completed DPF.
The profile of the corrugated surface produced should be selected
to facilitate the second phase.
[0291] Next the fluted material is fed in the x-direction into the
second mechansim. Options include having sharply creased or softly
bent flutes, and having the y-period of the corrugation greater
than, equal to, or less than the y-period of the DPF, and the
overall profile of the corrugation. Next the fluted material is
advanced in the x-direction relative to the folding mechansim. The
material may be fed into the mechansim, or the mechansim may be
passed across the material. This could be many devices, including
mating patterned rollers, designed by interpolating the folded
structures geometry into polar coordinates, as explained in
Sectionl 8.5, and computer controled articulating mechanisms. Two
features of the second phase is the no signifigant stretching of
the sheet material and its conversion from a fluted corugation type
geometry to the patterned folded structure. Remarkably, this
operation has been carried out successfully on paper and metal
foils.
[0292] In the descriptions above one may also use our Batch Process
with the roles of x and y interchanged.
[0293] 18.4 Description of Two Phase Process
[0294] The Two-Phase Process gives a procedure for producing facet
folded structures that is valuable for production. The usual
complications of folding the individual edges of a tessellation or
fold network, due to the interlocking effect of the fold vertices,
resulting in the simultaneous rotation and motion of facets, and
the overall contraction of the network in both the x and y
directions, had previously been the obstacle to mass-production.
This new process reduces the material flow problem to two phases,
each of which may readily be carried out by a variety of
mechanisms. This valuable simplifcation of the geometry of the
material flow is possible in part through the non-obvious dropping
of the linkage constraints and the non-obvious feasability of phase
2 of the process.
[0295] In the first phase the material is pre-gathered in the
y-direction, by imparting flutes or corrugations of various
profiles. The contraction ratio in the y-direction should be nearly
the same for the pre-gathered material and the folded structure,
and should be distributed evenly enough to facilitate phase 2. In
many cases the creases or bends of the profile will be removed in
phase 2, and optionally are not required to be centered on the
vertices of the folded structure.
[0296] In phase 2 the material is advanced continuously in the
x-direction relative to the contact area of the forming mechansim
of phase 2. This may be pairs of mating rollers, articulating
guides, or other forming tools. The contact tool continuously
advances in the xdirection across the material, trasforming the
flutes of the pregathered material into defined networks of folds.
The material moves locally in the ydirection, but the overall
contraction ratio in the y-direction of the material before and
after phase 2 does not change signifigantly. The forming process
does not signifigantly stretch the material.
[0297] In the descriptions above one may also use our Two-Phase
Process with the roles of x and y interchanged.
[0298] 18.5 Folding Rollers
[0299] The use of rollers to form patterned structures by
embossing, stamping, or crushing sheet materials is well
established. The use of rollers to fold sheet material into DPFs
and other folded networks is new. It is surprising that a
pre-gathered sheet material, such as a sine-wave corrugation, may
be manipulated in three-space under the influence of rollers into a
folded tessellation structure. The fact that fluted material will
convert into faceted material under a folding operation that does
not signifigantly stretch the material inside rollers is completely
non-obvious, and extremely valuable for mass production.
[0300] With the methodology for generating DPF vertex coordinates,
DPF forming rollers can also be readily designed by interpolating
the slab parallel to the XY plane containing the DPF onto a shell
of a cylindar. For two such rollers to fully engage, it is
sometimes necessary to expand the valleys while preserving the
ridges of the patterned rollers.
* * * * *