U.S. patent application number 11/296350 was filed with the patent office on 2006-06-22 for great-circle geodesic dome.
Invention is credited to Randell L. Mills.
Application Number | 20060135288 11/296350 |
Document ID | / |
Family ID | 36596733 |
Filed Date | 2006-06-22 |
United States Patent
Application |
20060135288 |
Kind Code |
A1 |
Mills; Randell L. |
June 22, 2006 |
Great-circle geodesic dome
Abstract
The present invention is a structural system and a method of
fabricating the system comprised of finite elements called basis
elements such as great circles or partial great circles to form a
great-circle geodesic sphere, dome, or arch called a great sphere,
dome, and arch, respectively. The structure is generated by forming
a pattern of the basis element rotated at incremental angles about
a rotational axis from an initial position to define a so-called
primary component orbitsphere-cvf. The primary component
orbitsphere defines a stationary pattern as well as a structural
element called a secondary component orbitsphere-cvf that is
constructed according to the pattern of the primary
orbitsphere-cvf. The secondary component orbitsphere-cvf is
initially oriented at the initial position and is incrementally
rotated about the rotational axis to form the great-sphere-type
structure.
Inventors: |
Mills; Randell L.;
(Princeton, NJ) |
Correspondence
Address: |
MANELLI DENISON & SELTER
2000 M STREET NW SUITE 700
WASHINGTON
DC
20036-3307
US
|
Family ID: |
36596733 |
Appl. No.: |
11/296350 |
Filed: |
December 8, 2005 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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60651006 |
Feb 9, 2005 |
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60647406 |
Jan 28, 2005 |
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60643149 |
Jan 13, 2005 |
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60637889 |
Dec 22, 2004 |
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Current U.S.
Class: |
473/379 ;
52/81.1 |
Current CPC
Class: |
E04B 1/3211 20130101;
E04B 7/105 20130101 |
Class at
Publication: |
473/379 ;
052/081.1 |
International
Class: |
A63B 37/12 20060101
A63B037/12; E04B 7/08 20060101 E04B007/08 |
Claims
1. A geodesic structure comprising: a plurality of circle elements
defining a geodesic structure, the circle elements having
substantially the same diameter bound to one another such that the
forces are equalized on the geodesic structure.
2. A geodesic structure according to claim 1, wherein the circle
elements are half circles and the geodesic structure comprises a
dome.
3. A geodesic structure according to claim 1, wherein the circle
elements are full circles and the geodesic structure comprises a
sphere.
4. A geodesic structure according to claim 1, wherein the geodesic
structure comprises an arch.
5. A geodesic structure according to claim 1, wherein no more than
two circle elements are aligned on an axis.
6. A geodesic structure according to claim 1, wherein the circle
elements are aligned according to Eqs. (1-71).
7. A geodesic structure according to claim 1, wherein the circle
elements all have the same diameter.
8. A geodesic structure according to claim 1, wherein the circle
elements overlap one another.
9. A geodesic structure according to claim 1, wherein the circle
elements are intertwined with one another.
10. A geodesic structure according to claim 1, wherein the circle
elements are arranged according to the structure of a primary
component orbitsphere-cvf wherein each circle element of the
primary component orbitsphere-cvf is replaced by a secondary
orbitsphere-cvf.
11. A geodesic structure according to claim 10, wherein each circle
element of each orbitsphere-cvf element comprises a structural
element.
12. A geodesic structure according to claim 11, wherein the
structural element comprises at least one a tube, bar, rod, or
beam.
13. A geodesic structure according to claim 12, wherein the
structural elements are welded, bolted, riveted, clamped, glued, or
otherwise fastened at their crossings with other such elements to
form the architectural structure.
14. A geodesic structure according to claim 1, wherein the
architectural structure has the form or a portion of the form
provided by at least one of Eqs. (67-69).
15. A geodesic structure according to claim 1, wherein the geodesic
structure comprises a container, dish, or vessel formed from the
circle elements.
16. A geodesic structure according to claim 1, wherein the circular
elements comprise framing to provide structural strength.
17. A geodesic structure according to claim 16, wherein the framing
is covered with a continuous membrane or a tiling that forms a
continuous or partially continuous covering of the framing.
18. A geodesic structure according to claim 10, wherein a number of
secondary component orbitsphere-cvf elements is equal to the number
of great-circle elements of the primary component
orbitsphere-cvf.
19. A uniform current-density structure comprising a plurality of
great-circle element loops wherein the circle elements are arranged
according to the structure of a primary component orbitsphere-cvf
wherein each circle element of the primary component
orbitsphere-cvf is replaced by a secondary orbitsphere-cvf to form
Y.sub.0.sup.0(.phi.,.theta.).
20. The uniform current-density function having a magnetic moment
along the z-axis that is twice the magnitude of the magnetic moment
in the xy-plane.
21. The geodesic structure according to claim 10, wherein the
circle elements are arranged according to the structure of a
primary component orbitsphere-cvf represented by [ x ' y ' z ' ] =
[ 1 2 + cos .times. .times. .theta. 2 1 2 - cos .times. .times.
.theta. 2 - sin .times. .times. .theta. 2 1 2 - cos .times. .times.
.theta. 2 1 2 + cos .times. .times. .theta. 2 sin .times. .times.
.theta. 2 sin .times. .times. .theta. 2 - sin .times. .times.
.theta. 2 cos .times. .times. .theta. ] .function. [ 0 r n .times.
cos .times. .times. .PHI. r n .times. sin .times. .times. .PHI. ]
.times. [ x ' y ' z ' ] = [ ( 1 2 - cos .times. .times. .theta. 2 )
.times. r n .times. cos .times. .times. .PHI. - sin .times. .times.
.theta. 2 .times. r n .times. sin .times. .times. .PHI. ( 1 2 + cos
.times. .times. .theta. 2 ) .times. r n .times. cos .times. .times.
.PHI. + sin .times. .times. .theta. 2 .times. r n .times. sin
.times. .times. .PHI. - sin .times. .times. .theta. 2 .times. r n
.times. cos .times. .times. .PHI. + cos .times. .times. .theta.
.times. .times. r n .times. sin .times. .times. .PHI. ]
##EQU96##
22. The geodesic structure according to claim 21, wherein the
circle elements are arranged according to the structure of a
primary component orbitsphere-cvf wherein each circle element of
the primary component orbitsphere-cvf is replaced by a secondary
orbitsphere-cvf represented by [ x ' y ' z ' ] = m = 1 m = M
.times. [ 1 2 + cos .function. ( m .times. .times. 2 .times. .pi. M
) 2 1 2 - cos .function. ( m .times. .times. 2 .times. .pi. M ) 2 -
sin .function. ( m .times. .times. 2 .times. .pi. M ) 2 1 2 - cos
.function. ( m .times. .times. 2 .times. .pi. M ) 2 1 2 + cos
.function. ( m .times. .times. 2 .times. .pi. M ) 2 sin .function.
( m .times. .times. 2 .times. .pi. M ) 2 sin .function. ( m .times.
.times. 2 .times. .pi. M ) 2 - sin .function. ( m .times. .times. 2
.times. .pi. M ) 2 cos .function. ( m .times. .times. 2 .times.
.pi. M ) ] n = 1 n = N .times. [ cos .function. ( .pi. 4 ) - sin
.function. ( .pi. 4 ) 0 sin .function. ( .pi. 4 ) .times. cos
.times. .times. ( n .times. .times. 2 .times. .pi. N ) cos
.function. ( .pi. 4 ) .times. cos .times. .times. ( n .times.
.times. 2 .times. .pi. N ) sin .times. .times. ( n .times. .times.
2 .times. .pi. N ) - sin .function. ( .pi. 4 ) .times. sin .times.
.times. ( n .times. .times. 2 .times. .pi. N ) - cos .function. (
.pi. 4 ) .times. sin .times. .times. ( n .times. .times. 2 .times.
.pi. N ) cos .times. .times. ( n .times. .times. 2 .times. .pi. N )
] .function. [ 0 r n .times. cos .times. .times. .PHI. r n .times.
sin .times. .times. .PHI. ] ##EQU97##
23. The uniform current-density structure of claim 20 wherein the
circle elements are arranged according to the structure of a
primary component orbitsphere-cvf represented by [ x ' y ' z ' ] =
[ sin 2 .times. .theta. 2 - cos 2 .times. .theta. 2 - sin .times.
.times. .theta. 2 - cos 2 .times. .theta. 2 sin 2 .times. .theta. 2
- sin .times. .times. .theta. 2 sin .times. .times. .theta. 2 sin
.times. .times. .theta. 2 - cos .times. .times. .theta. ]
.function. [ 0 r n .times. cos .times. .times. .PHI. r n .times.
sin .times. .times. .PHI. ] ( 36 ) [ x ' y ' z ' ] = [ - cos 2
.times. .theta. 2 .times. r n .times. cos .times. .times. .PHI. -
sin .times. .times. .theta. 2 .times. r n .times. sin .times.
.times. .PHI. sin 2 .times. .theta. 2 .times. r n .times. cos
.times. .times. .PHI. - sin .times. .times. .theta. 2 .times. r n
.times. sin .times. .times. .PHI. sin .times. .times. .theta. 2
.times. r n .times. cos .times. .times. .PHI. - cos .times. .times.
.theta. .times. .times. r n .times. sin .times. .times. .PHI. ] (
37 ) ##EQU98##
24. The uniform current-density structure of claim 23 wherein the
circle elements are arranged according to the structure of a
primary component orbitsphere-cvf wherein each circle element of
the primary component orbitsphere-cvf is replaced by a secondary
orbitsphere-cvf represented by [ x ' y ' z ' ] = m = 1 m = M
.times. [ 1 2 + cos .function. ( m .times. .times. 2 .times. .pi. M
) 2 1 2 - cos .function. ( m .times. .times. 2 .times. .pi. M ) 2 -
sin .function. ( m .times. .times. 2 .times. .pi. M ) 2 1 2 - cos
.function. ( m .times. .times. 2 .times. .pi. M ) 2 1 2 + cos
.function. ( m .times. .times. 2 .times. .pi. M ) 2 sin .function.
( m .times. .times. 2 .times. .pi. M ) 2 sin .function. ( m .times.
.times. 2 .times. .pi. M ) 2 - sin .function. ( m .times. .times. 2
.times. .pi. M ) 2 cos .function. ( m .times. .times. 2 .times.
.pi. M ) ] n = 1 n = N .times. [ cos .function. ( .pi. 4 ) - sin
.function. ( .pi. 4 ) 0 sin .function. ( .pi. 4 ) .times. cos
.function. ( n .times. .times. 2 .times. .pi. N ) cos .function. (
.pi. 4 ) .times. cos .function. ( n .times. .times. 2 .times. .pi.
N ) sin .function. ( n .times. .times. 2 .times. .pi. N ) - sin
.function. ( .pi. 4 ) .times. sin .function. ( n .times. .times. 2
.times. .pi. N ) - cos .function. ( .pi. 4 ) .times. sin .function.
( n .times. .times. 2 .times. .pi. N ) cos .function. ( n .times.
.times. 2 .times. .pi. N ) ] .function. [ 0 r n .times. cos .times.
.times. .PHI. r n .times. sin .times. .times. .PHI. ] ##EQU99##
Description
[0001] This application claims priority U.S. Provisional Patent
Application Ser. No.: 60/651,006, filed 9 Feb. 2005; 60/647,406,
filed 28 Jan. 2005; 60/643,149, field 13 Jan. 2005; and 60/637,889,
filed 22 Dec. 2004, the complete disclosures of which are
incorporated herein by reference.
FIELD OF THE INVENTION
[0002] A conventional dome comprising a saddle structure or a
conventional geodesic dome made of triangular basis elements have
elements under tension; whereas, a perfect dome is essentially or
entirely under compression since all of weight is equally
distributed. Furthermore, a perfect sphere has no unique position
on the surface or unique axis. This invention relates to a
structure that more closely replicates a perfect sphere or dome
using finite elements derived from great circles. In an embodiment,
the surface has a characteristic that axes defined by crossings of
great-circle elements are not unique. Such an architectural
structure or container is anticipated to be stronger and provide
more efficiency of material requirements than in conventional
systems such as a structure made of triangular basis elements as in
the case of a conventional geodesic dome. The structure is also
more esthetic and natural. It further permits the construction of
near perfect spherical dishes without machining out a solid
material which is a very challenging fabrication problem. Broad
application in architecture and industry by one skilled in the Art
is anticipated and within the scope of the current Invention.
BACKGROUND OF THE INVENTION
[0003] This Invention relates to a geometrical derivation of a
means to generate a perfect sphere using great circles. The
disclosure and background are given in the book, R. Mills, The
Grand Unified Theory of Classical Quantum Mechanics, January 2006
Edition, BlackLight Power, Inc., Cranbury, N.J.; posted at
http://www.blacklightpower.com/bookdownload.shtml ("Mills GUT"),
which is incorporated by reference in its entirety. The Chapter One
and Appendix III are preferred references which are also
incorporated by this reference.
SUMMARY OF THE INVENTION
[0004] The object of this invention is to provide a sphere or
spherical section such as a dome as a universal architectural
structure or container, that is a more ideal of a perfect sphere or
dome or spherical section using finite elements.
[0005] The present invention is a structural system and a method of
fabricating the system comprised of finite elements called basis
elements such as great circles or partial great circles to form a
great-circle geodesic sphere, dome, or arch called a great sphere,
dome, and arch, respectively. The structure is generated by forming
a pattern of the basis element rotated at incremental angles about
a rotational axis from an initial position to define a so-called
primary component orbitsphere-cvf. The primary component
orbitsphere defines a stationary pattern as well as a structural
element called a secondary component orbitsphere-cvf that is
constructed according to the pattern of the primary
orbitsphere-cvf. The secondary component orbitsphere-cvf is
initially oriented at the initial position and is incrementally
rotated about the rotational axis to form the great-sphere-type
structure. The structure defined as Y.sub.0.sup.0(.phi.,.theta.)
may formed according to Eqs. (67-71) of the ANALYTICAL EQUATIONS TO
GENERATE THE ORBITSPHERE CURRENT VECTOR FIELD AND THE UNIFORM
CURRENT (CHARGE)-DENSITY FUNCTION Y.sub.0.sup.0(.phi.,.theta.)
section.
[0006] Another embodiment of the Invention comprises a uniform
current density function on a two dimensional surface defined as
Y.sub.0.sup.0(.phi.,.theta.) and a method and constructing
Y.sub.0.sup.0(.phi.,.theta.). The current density function may have
a defined angular momentum along two orthogonal axes that is
determined by the selection of the desired angular momentum of the
basis elements and by the selection the rotational matrices that
form the Y.sub.0.sup.0(.phi.,.theta.) with the desired angular
momentum projections. The angular momentum components have
corresponding magnetic moments in the embodiment wherein the
elements are current loops. A further embodiment comprises a
uniform mass-density structure with desired angular momentum
components. This embodiment is constructed by using mass-flowing
elements rather than current elements.
BRIEF DESCRIPTION OF THE DRAWINGS
[0007] FIG. 1 is the element pattern given by Eq. (1) shown with 6
degree increments of .theta. from the perspective of looking along
the z-axis. The great circle element that served as a basis element
that was initially in the yz-plane is shown as red in accordance
with the present invention;
[0008] FIG. 2 is the element pattern of the orbitsphere-cvf
component of STEP ONE shown with 6 degree increments of .theta.
from the perspective of looking along the z-axis. The yz-plane
great circle element that served as a basis element that was
initially in the yz-plane is shown as red in accordance with the
present invention;
[0009] FIG. 3 is the element pattern of the orbitsphere-cvf
component of STEP ONE shown with 6 degree increments of .theta.
from the perspective of looking along the z-axis. The great circle
element that served as a basis element that was initially in the
xz-plane is shown as red in accordance with the present
invention;
[0010] FIG. 4 the element pattern given by Eq. (7) shown with 6
degree increments of .theta. from the perspective of looking along
the z-axis. The great circle element that served as a basis element
that was initially in the xy-plane is shown as red is in accordance
with the present invention;
[0011] FIG. 5 is the element pattern for the rotation of the
xy-plane great circle about the (-i.sub.x,0i.sub.y,i.sub.z)-axis
(Eq. (10)) shown with 6 degree increments of .theta. from the
perspective of looking along the z-axis. The great circle element
that served as a basis element that was initially in the xy-plane
is shown as red in accordance with the present invention;
[0012] FIG. 6 is the element pattern of the orbitsphere-cvf
component of STEP TWO shown with 6 degree increments of .theta.
from the perspective of looking along the z-axis. The great circle
element that served as a basis element that was initially in the
xy-plane is shown as red in accordance with the present
invention;
[0013] FIG. 7 is the element pattern of the orbitsphere-cvf
component of STEP TWO shown with 6 degree increments of .theta.
from the perspective of looking along the z-axis. The great circle
element that served as a basis element that was initially in the
yz-plane is shown as red in accordance with the present
invention;
[0014] FIG. 8 is the element pattern of the orbitsphere-cvf shown
with 6 degree increments of .theta. from the perspective of looking
along the z-axis onto which L.sub.R, the resultant vector of the
L.sub.xy and L.sub.z components, was aligned in accordance with the
present invention;
[0015] FIG. 9 is the schematic of the relative dimensions of the
component orbitsphere-cvfs (STEP-ONE component shown in blue and
STEP-TWO component shown in red) that make-up the orbitsphere-cvf
in accordance with the present invention;
[0016] FIG. 10 is the element pattern given by Eq. (30) shown with
6 degree increments of .theta. from the perspective of looking
along the z-axis. The great circle element that served as a basis
element that was initially in the yz-plane is shown as red in
accordance with the present invention;
[0017] FIG. 11 is the element pattern of the orbitsphere-cvf
component given by Eq. (33) that is orthogonal to that of STEP ONE
shown with 6 degree increments of .theta. from the perspective of
looking along the z-axis. The yz-plane great circle element that
served as a basis element that was initially in the yz-plane is
shown as red in accordance with the present invention;
[0018] FIG. 12 is the element pattern given by Eq. (37) shown with
6 degree increments of .theta. from the perspective of looking
along the z-axis obtained from Eq. (32) by rotation about the
(i.sub.x,-i.sup.y,0i.sub.z)-axis by .pi. using Eq. (34). The great
circle element that served as a basis element that was initially in
the yz-plane is shown as red in accordance with the present
invention;
[0019] FIG. 13 is the orbitsphere, a two dimensional spherical
shell in accordance with the present invention;
[0020] FIG. 14 is a representation of the uniform element pattern
of the Y.sub.0.sup.0(.phi.,.theta.) orbitsphere shown with 30
degree increments (N=M=12 in Eq. (67)) of the angle to generate the
orbitsphere current-vector field corresponding to Eq. (30) and 30
degree increments of the rotation of this basis element about the
(i.sub.x,i.sub.y,0i.sub.z)-axis corresponding to Eq. (4). The
perspective is along the z-axis. The great circle element that
served as a basis element that was initially in the plane along the
(i.sub.x,-i.sub.y,0i.sub.z)- and z-axes of each secondary component
orbitsphere-cvf is shown as red. Note that it is stationary over
the convolution due to phase matching in accordance with the
present invention;
[0021] FIG. 15 is a representation of the uniform element pattern
of the Y.sub.0.sup.0(.phi.,.theta.) orbitsphere shown with 30
degree increments (N=M=12 in Eq. (67)) of the angle to generate the
orbitsphere current-vector field corresponding to Eq. (30) and 30
degree increments of the rotation of this basis element about the
(i.sub.x,i.sub.y,0i.sub.z)-axis corresponding to Eq. (4). The great
circle element that served as a basis element that was initially in
the plane along the (i.sub.x,-i.sub.y,0i.sub.z)- and z-axes of each
secondary component orbitsphere-cvf is shown as red. The
perspective is transverse to the z-axis in accordance with the
present invention;
[0022] FIG. 16 is a representation of the uniform element pattern
of the Y.sub.0.sup.0(.phi.,.theta.) orbitsphere shown with 30
degree increments (N=M=12 in Eq. (68)) of the angle to generate the
orbitsphere current-vector field corresponding to Eq. (32) and 30
degree increments of the rotation of this basis element about the (
- 1 2 .times. i x , 1 2 .times. i y , i z ) ##EQU1## -axis
corresponding to Eq. (20). The great circle current loop that
served as a basis element that was initially in the yz-plane of
each secondary component orbitsphere-cvf is shown as red. Note that
it is not stationary over the convolution due to phase matching. It
is out of phase with the secondary component orbitsphere by a
-.pi./4 rotation about the z-axis. The perspective is along the
z-axis in accordance with the present invention;
[0023] FIG. 17 is a representation of the uniform element pattern
of the Y.sub.0.sup.0(.phi.,.theta.) orbitsphere shown with 30
degree increments (N=M=12 in Eq. (68)) of the angle to generate the
orbitsphere current-vector field corresponding to Eq. (32) and 30
degree increments of the rotation of this basis element about the (
- 1 2 .times. i x , 1 2 .times. i y , i z ) ##EQU2## -axis
corresponding to Eq. (20). The great circle current loop that
served as a basis element that was initially in the yz-plane of
each secondary component orbitsphere-cvf is shown as red. The
perspective is transverse to the z-axis in accordance with the
present invention;
[0024] FIG. 18 is a representation of the uniform element pattern
of the Y.sub.0.sup.0(.phi.,.theta.) orbitsphere shown with 30
degree increments (N=M=12 in Eq. (68)) of the angle to generate the
orbitsphere current-vector field corresponding to Eq. (32) and 30
degree increments of the rotation of this basis element about the (
- 1 2 .times. i x , 1 2 .times. i y , i z ) ##EQU3## -axis
corresponding to Eq. (20). The great circle current loop that
served as a basis element that was initially in the yz-plane of
each secondary component orbitsphere-cvf is not highlighted. The
perspective is along the z-axis in accordance with the present
invention;
[0025] FIG. 19 is a representation of the uniform current pattern
of the Y.sub.0.sup.0(0,O) orbitsphere shown with 30 degree
increments (N=M=12 in Eq. (71)) of the angle to generate the
orbitsphere current-vector field corresponding to Eq. (70) and 30
degree increments of the rotation of this basis element about the (
- 1 2 .times. i x , 1 2 .times. i y , i z ) ##EQU4## -axis
corresponding to Eq. (20). The great circle current loop that
served as a basis element that was initially in the plane along the
( - 1 2 .times. i x , 1 2 .times. i y , i z ) ##EQU5## --and y-axes
of each secondary component orbitsphere-cvf is shown as red. Note
that it is not stationary over the convolution due to phase
matching. It is out of phase with the secondary component
orbitsphere by -.pi./4 about the y-axis. The perspective is along
the z-axis in accordance with the present invention;
[0026] FIG. 20 is a representation of the uniform current pattern
of the Y.sub.0.sup.0(.phi.,.theta.) orbitsphere shown with 30
degree increments (N=M=12 in Eq. (71)) of the angle to generate the
orbitsphere current-vector field corresponding to Eq. (70) and 30
degree increments of the rotation of this basis element about the (
- 1 2 .times. i x , 1 2 .times. i y , i z ) ##EQU6## -axis
corresponding to Eq. (20). The great circle current loop that
served as a basis element that was initially in the plane along the
( - 1 2 .times. i x , 1 2 .times. i y , i z ) ##EQU7## --and y-axes
of each secondary component orbitsphere-cvf is shown as red. The
perspective is transverse to the z-axis in accordance with the
present invention, and
[0027] FIG. 21 is a representation of the uniform current pattern
of the Y.sub.0.sup.0(.phi.,.theta.) orbitsphere shown with 30
degree increments (N=M=12 in Eq. (71)) of the angle to generate the
orbitsphere current-vector field corresponding to Eq. (70) and 30
degree increments of the rotation of this basis element about the (
- 1 2 .times. i x , 1 2 .times. i y , i z ) ##EQU8## -axis
corresponding to Eq. (20). The great circle current loop that
served as a basis element that was initially in the plane along the
( - 1 2 .times. i x , 1 2 .times. i y , i z ) ##EQU9## --and y-axes
of each secondary component orbitsphere-cvf is not highlighted. The
perspective is along the z-axis in accordance with the present
invention.
DETAILED DESCRIPTION OF THE INVENTION
[0028] An embodiment of the present Invention comprises an
architectural system and a method of fabricating the architectural
system. The system comprises a structure formed from great-circle
elements or sections of such elements to form a sphere or a section
of a sphere such as a half shell, a dome, or an arch. The elements
may have essentially the same diameter and may be bound together at
the crossings of the elements or may be intertwined at the
crossings. Ideally the system is constructed to equalize the forces
throughout the surface of the structure. Preferably, the structure
is two-dimensional with equal weight or pressure distribution.
[0029] In an embodiment, the elements are full great circles of the
same diameter, and the corresponding structure is a sphere. In
another embodiment, the basis elements are half great circles of
the same radius, and the corresponding structure is a dome.
[0030] In an embodiment, the architectural structure has the form
given by at least one of Eqs. (67-71) of the Analytical Equations
to Generate the Orbitsphere Current Vector Field and the Uniform
Current (Charge)-Density Function Y.sub.0.sup.0(.phi.,.theta.)
section. In an embodiment, the elements are arranged according to
the structure of a primary component orbitsphere-cvf wherein each
great-circle element of the primary component orbitsphere-cvf is
replaced by a secondary orbitsphere-cvf wherein each great-circle
element of each orbitsphere-cvf element comprises a structural
element such as a tube, bar, rod, beam, or similar element that is
in the form of a great circle or a section of a great circle.
[0031] In an embodiment, the curved elements such as great circles
or partial sections of great circles are approximated with straight
subelements. In an embodiment, an element having an average
structure of a curve can be constructed from elements that are not
curved such as straight elements. Other such subelements are known
to those skilled in the Art.
[0032] In any embodiment, the great-circle basis elements may
comprise the framing to provide structural strength, and the
framing may be covered with a continuous membrane or a tiling that
forms a continuous or partially continuous covering of the framing.
In other embodiments, the structures may accommodate structures
such as doors and windows and other structural elements.
[0033] In an embodiment, the number of such secondary component
orbitsphere-cvf elements is equal to the number of great-circle
elements of the primary component orbitsphere-cvf. However, the
number of elements of the secondary component orbitsphere, N, and
the primary component orbitsphere, M, do not need be the same. One
skilled in the Art could also determine the optimum numbers N and M
to achieve a desired goal such as least cost to achieve an
application with the desired performance characteristics.
[0034] In an embodiment the angles N and m of Eqs. (67), (68), and
(71) are integer multiples within the range of .+-.0.00001% to 50%,
preferably within the range of .+-.0.0001% to 25%; more preferably
.+-.0.01% to 10%, and most preferably .+-.1% to 5%. The higher the
number of elements, N and M, the more closely the structure
approximates a perfect sphere or partial sphere corresponding to
great circle elements or partial great circle elements,
respectively. The tolerance from a perfect sphere or partial sphere
depends on the particular application. One skilled in the Art would
be able to determine the structure and geometrical comprise between
the number of elements and realization of perfect weight or
pressure distribution, or physical form.
[0035] In an embodiment, the angles n2.pi./N and m2.pi./M of Eqs.
(67), (68), and (71) are generally represented by n.theta..sub.1/N
and m.theta..sub.2/M wherein each of .theta..sub.1 and
.theta..sub.2 is an independent angle within the range of
0.ltoreq..theta..sub.1,2.ltoreq.2.pi.. One skilled in the Art could
select the range to form the desired structure from the basis
element such as a great circle or partial great circle.
[0036] The architectural system may be used as a building or part
of a building such as an arena, stadium, atrium, roof of a
structure, a dwelling, or a structural component of a structure
such as a support for a bridge or building.
[0037] In other embodiments, the invention comprises a container,
dish, or vessel formed from great-circle elements or sections of
such elements to form a sphere or a section of a sphere such as a
half shell, a dome, or an arch. The container may be for solids,
liquids, or gases. Spherical dishes are also embodiments. A
spherical antenna or mirror for uses such as communications or
telescopes are further embodiments. In another embodiment, the
structure is used as the fuselage or structural elements of a
submarine or pressurized gas container for applications such as
transporting liquefied natural gas.
[0038] The system may be formed directly from great-circle elements
or sections of great-circle elements. Alternatively, a number of
secondary component orbitsphere-cvf elements or sections of such
elements may be fabricated and these elements may be connected to
form the structure. The great-circle basis elements may be welded,
bolted, riveted, clamped, glued, or otherwise fastened at their
crossings with other such elements to form the architectural
structure.
[0039] Another embodiment of the Invention comprises a uniform
current density function on a two dimensional surface defined as
Y.sub.0.sup.0(.phi.,.theta.) and a method and constructing
Y.sub.0.sup.0(.phi.,.theta.). The current density function may have
a defined angular momentum along two orthogonal axes that is
determined by the selection of the desired angular momentum of the
basis elements and by the selection the rotational matrices that
form the Y.sub.0.sup.0(.phi.,.theta.) with the desired angular
momentum projections. The angular momentum components have
corresponding magnetic moments in the embodiment wherein the
elements are current loops. A further embodiment comprises a
uniform mass-density structure with desired angular momentum
components. This embodiment is constructed by using mass-flowing
elements rather than current elements.
[0040] Embodiments of the structures and methods are given in the
ANALYTICAL EQUATIONS TO GENERATE THE ORBITSPHERE CURRENT VECTOR
FIELD AND THE UNIFORM CURRENT (CHARGE)-DENSITY FUNCTION
Y.sub.0.sup.0(.phi.,.theta.) section. A specific embodiment to
construct a uniform current density function with L xy = 4 .times.
and .times. .times. L z = 2 ##EQU10## is given. Other angular
momentum components are within the scope of the Invention with the
use of basis elements of the corresponding angular momentum as
would be used by one skilled in the Art. The great dome embodiment
is also taught in this section with the use of structural elements
rather than current-loop basis elements.
[0041] Analytical Equations to Generate the Orbitsphere Current
Vector Field and the Uniform Current (Charge)-Density Function
Y.sub.0.sup.0(.theta.,.phi.)
[0042] STEP ONE by the Rotation of a Great Circle About the
(i.sub.x,i.sub.y,0i.sub.z)-Axis by 2.pi.
[0043] Great Circle in the yz-Plane About the
(i.sub.x,i.sub.y,0i.sub.z)-Axis
[0044] Following the procedure given in Fowles [1], the
orbitsphere-cvf component of STEP ONE is generated by the rotation
of a great circle in the yz-plane about the
(i.sub.x,i.sub.y,0i.sub.z)-axis by 2.pi.. A first transformation
matrix is generated by the combined rotation of a great circle in
the yz-plane about the z-axis by .pi./4 then about the x-axis by
.theta. where positive rotations about an axis are defined as
clockwise: [ x ' y ' z ' ] = [ cos .times. .times. ( .pi. 4 ) sin
.times. .times. ( .pi. 4 ) .times. 0 - sin .times. .times. ( .pi. 4
) .times. .times. cos .times. .times. .theta. cos .times. .times. (
.pi. 4 ) .times. .times. cos .times. .times. .theta. sin .times.
.times. .theta. sin .times. .times. ( .pi. 4 ) .times. .times. sin
.times. .times. .theta. - cos .times. .times. ( .pi. 4 ) .times.
.times. sin .times. .times. .theta. cos .times. .times. .theta. ]
.function. [ 0 r n .times. cos .times. .times. .PHI. r n .times.
sin .times. .times. .PHI. ] ( 1 ) ##EQU11##
[0045] The transformation matrix about (i.sub.x,i.sub.y,0i.sub.z)
is given by multiplication of the output of the matrix given by Eq.
(1) by the matrix corresponding to a rotation about the z-axis of
-.pi./4. The output of the matrix given by Eq. (1) is shown in FIG.
1 wherein .theta. is varied from 0 to 2.pi.. The rotation matrix
about the z-axis by -.pi./4, zrot .times. .times. ( - .pi. 4 ) ,
##EQU12## is given by zrot .times. .times. ( - .pi. 4 ) = [ cos
.times. .times. ( .pi. 4 ) - sin .times. .times. ( .pi. 4 ) 0 sin
.times. .times. ( .pi. 4 ) cos .times. .times. ( .pi. 4 ) 0 0 0 1 ]
.times. .times. Thus , ( 2 ) [ x ' y ' z ' ] = .times. zrot .times.
.times. ( - .pi. 4 ) .times. [ cos .times. .times. ( .pi. 4 ) sin
.times. .times. ( .pi. 4 ) .times. 0 - sin .times. .times. ( .pi. 4
) .times. .times. cos .times. .times. .theta. cos .times. .times. (
.pi. 4 ) .times. .times. cos .times. .times. .theta. sin .times.
.times. .theta. sin .times. .times. ( .pi. 4 ) .times. .times. sin
.times. .times. .theta. - cos .times. .times. ( .pi. 4 ) .times.
.times. sin .times. .times. .theta. cos .times. .times. .theta. ]
.function. [ 0 r n .times. cos .times. .times. .PHI. r n .times.
sin .times. .times. .PHI. ] ( 3 ) ##EQU13##
[0046] Substitution of the matrix given by Eq. (2) into Eq. (3)
gives [ x ' y ' z ' ] = [ 1 2 + cos .times. .times. .theta. 2 1 2 -
cos .times. .times. .theta. 2 - sin .times. .times. .theta. 2 1 2 -
cos .times. .times. .theta. 2 1 2 + cos .times. .times. .theta. 2
sin .times. .times. .theta. 2 sin .times. .times. .theta. 2 - sin
.times. .times. .theta. 2 cos .times. .times. .theta. ] .function.
[ 0 r n .times. cos .times. .times. .PHI. r n .times. sin .times.
.times. .PHI. ] ( 4 ) [ x ' y ' z ' ] = [ ( 1 2 - cos .times.
.times. .theta. 2 ) .times. .times. r n .times. cos .times. .times.
.PHI. - sin .times. .times. .theta. 2 .times. r n .times. sin
.times. .times. .PHI. ( 1 2 + cos .times. .times. .theta. 2 )
.times. .times. r n .times. cos .times. .times. .PHI. + sin .times.
.times. .theta. 2 .times. r n .times. sin .times. .times. .PHI. -
sin .times. .times. .theta. 2 .times. r n .times. cos .times.
.times. .PHI. + cos .times. .times. .theta. .times. .times. r n
.times. sin .times. .times. .PHI. ] ( 5 ) ##EQU14##
[0047] The orbitsphere-cvf component of STEP ONE that is generated
by the rotation of a great circle in the yz-plane about the
(i.sub.x,i.sub.y,0i.sub.z)-axis by 2.pi. corresponding to the
output of the matrix given by Eq. (5) is shown in FIG. 2 wherein
the sign of .phi. is positive for 0.ltoreq..theta..ltoreq..pi. or
and negative for .pi..ltoreq..theta..ltoreq.2.pi. in order to give
the angular momentum projections given in the Orbitsphere Equation
of Motion for l=0 section of Mills GUT,
[0048] Great Circle in the xz-Plane About the
(i.sub.x,i.sub.y,0i.sub.z)-Axis
[0049] Alternatively, STEP ONE comprises the rotation of a great
circle in the xz-plane about the (i.sub.x,i.sub.y,0i.sub.z)-axis by
2.pi.. The coordinates of the great circle are given by the matrix:
[ x ' y ' z ' ] = [ r n .times. cos .times. .times. .PHI. 0 r n
.times. sin .times. .times. .PHI. ] ( 6 ) ##EQU15##
[0050] The matrix for the rotation about the
(i.sub.x,i.sub.y,0i.sub.z)-axis is given by Eq. (4) wherein .theta.
is varied from 0 to 2.pi. and the sign of .phi. is positive for
0.ltoreq..theta..ltoreq..pi. and negative for
.pi..ltoreq..theta..ltoreq.2.pi. in order to give the angular
momentum projections given in the Orbitsphere Equation of Motion
for l=0 section. The current pattern that is equivalent to that
shown in FIG. 2 is shown in FIG. 3.
[0051] It follows from the results shown in FIGS. 2 and 3 that the
component orbitsphere-cvf for STEP ONE can further be generated by
the rotation of the linear combination of the basis-element great
circles in the yz- and xz-planes about the
(i.sub.x,i.sub.y,0i.sub.z)-axis by .pi. using Eqs. (4) and (6)
wherein .theta. is varied from 0 to .pi.. It is a general feature
for the generation of the components given in this Appendix that a
linear combination of the orthogonal basis-element great circles
can used rather than a single element wherein the range of .theta.
is varied from 0 to .pi. rather than from 0 to 2.pi..
[0052] STEP TWO by the Rotation of a Great Circle About the
(-i.sub.x,0i.sub.y,i.sub.z)-Axis by 2.pi. Followed by a Rotation
About the z-Axis by .pi./4
[0053] Great Circle in the xy-Plane About the
(-i.sub.x,0i.sub.y,i.sub.z)-Axis by 2.pi. Followed by a Rotation
About the z-Axis by .pi./4
[0054] Following the procedure given in Fowles [1], the
orbitsphere-cvf component of STEP TWO is generated by the rotation
of a great circle in the xy-plane about the
(-i.sub.x,0i.sub.y,i.sub.z)-axis by 2.pi. followed by a rotation
about the z-axis by .pi./4. A first transformation matrix is
generated by the combined rotation of a great circle in the
xy-plane about the y-axis by -.pi./4 then about the z-axis by
.theta.; [ x ' y ' z ' ] = [ cos .function. ( .pi. 4 ) .times. cos
.times. .times. .theta. sin .times. .times. .theta. sin .function.
( .pi. 4 ) .times. cos .times. .times. .theta. - cos .function. (
.pi. 4 ) .times. sin .times. .times. .theta. cos .times. .times.
.theta. - sin .function. ( .pi. 4 ) .times. sin .times. .times.
.theta. - sin .function. ( .pi. 4 ) 0 cos .function. ( .pi. 4 ) ]
.function. [ r n .times. cos .times. .times. .PHI. r n .times. sin
.times. .times. .PHI. 0 ] ( 7 ) ##EQU16##
[0055] The transformation matrix about (-i.sub.x,0i.sub.y,i.sub.z)
is given by multiplication of the output of the matrix given by Eq.
(7) by the matrix corresponding to a rotation about the y-axis of
.pi./4. The output of the matrix given by Eq. (7) is shown in FIG.
4 wherein .theta. is varied from 0 to 2.pi..
[0056] The rotation matrix about the y-axis by .pi./4, yrot
.function. ( .pi. 4 ) , ##EQU17## is given by yrot .function. (
.pi. 4 ) = [ cos .function. ( .pi. 4 ) 0 - sin .function. ( .pi. 4
) 0 1 0 sin .function. ( .pi. 4 ) 0 cos .function. ( .pi. 4 ) ]
.times. .times. Thus , ( 8 ) [ x ' y ' z ' ] = yrot .function. (
.pi. 4 ) [ cos .function. ( .pi. 4 ) .times. cos .times. .times.
.theta. sin .times. .times. .theta. sin .function. ( .pi. 4 )
.times. cos .times. .times. .theta. - cos .function. ( .pi. 4 )
.times. sin .times. .times. .theta. cos .times. .times. .theta. -
sin .function. ( .pi. 4 ) .times. sin .times. .times. .theta. - sin
.function. ( .pi. 4 ) 0 cos .function. ( .pi. 4 ) ] .function. [ r
n .times. cos .times. .times. .PHI. r n .times. sin .times. .times.
.PHI. 0 ] ( 9 ) ##EQU18##
[0057] Substitution of the matrix given by Eq. (8) into Eq. (9)
gives the current pattern for the rotation of the xy-plane great
circle about the (-i.sub.x,0i.sub.y,i.sub.z)-axis as shown in FIG.
5: [ x ' y ' z ' ] = [ 1 2 + cos .times. .times. .theta. 2 sin
.times. .times. .theta. 2 - 1 2 + cos .times. .times. .theta. 2 -
sin .times. .times. .theta. 2 cos .times. .times. .theta. - sin
.times. .times. .theta. 2 - 1 2 + cos .times. .times. .theta. 2 sin
.times. .times. .theta. 2 1 2 + cos .times. .times. .theta. 2 ]
.function. [ r n .times. cos .times. .times. .PHI. r n .times. sin
.times. .times. .PHI. 0 ] ( 10 ) ##EQU19##
[0058] STEP TWO is then given by a rotation of this result about
z-axis by .pi./4 using zrot .function. ( .pi. 4 ) ##EQU20## given
by zrot .function. ( .pi. 4 ) = [ cos .function. ( .pi. 4 ) sin
.function. ( .pi. 4 ) 0 - sin .function. ( .pi. 4 ) cos .function.
( .pi. 4 ) 0 0 0 1 ] ( 11 ) ##EQU21##
[0059] Thus, STEP TWO is given by [ x ' y ' z ' ] = zrot .function.
( .pi. 4 ) yrot .function. ( .pi. 4 ) [ cos .function. ( .pi. 4 )
.times. cos .times. .times. .theta. sin .times. .times. .theta. sin
.function. ( .pi. 4 ) .times. cos .times. .times. .theta. - cos
.function. ( .pi. 4 ) .times. sin .times. .times. .theta. cos
.times. .times. .theta. - sin .function. ( .pi. 4 ) .times. sin
.times. .times. .theta. - sin .function. ( .pi. 4 ) 0 cos
.function. ( .pi. 4 ) ] .function. [ r n .times. cos .times.
.times. .PHI. r n .times. sin .times. .times. .PHI. 0 ] ( 12 )
##EQU22##
[0060] Substitution of the matrices given by Eqs. (8) and (11) into
Eq. (12) gives [ x ' y ' z ' ] = [ 1 2 + cos .times. .times.
.theta. 2 2 - sin .times. .times. .theta. 2 cos .times. .times.
.theta. 2 + sin .times. .times. .theta. 2 - 1 2 + cos .times.
.times. .theta. 2 2 - sin .times. .times. .theta. 2 - 1 2 + cos
.times. .times. .theta. 2 2 - sin .times. .times. .theta. 2 cos
.times. .times. .theta. 2 - sin .times. .times. .theta. 2 - 1 2 +
cos .times. .times. .theta. 2 2 - sin .times. .times. .theta. 2 - 1
2 + cos .times. .times. .theta. 2 sin .times. .times. .theta. 2 1 2
+ cos .times. .times. .theta. 2 ] .times. [ r n .times. cos .times.
.times. .PHI. r n .times. sin .times. .times. .PHI. 0 ] ( 13 ) [ x
' y ' z ' ] = [ ( 1 2 + cos .times. .times. .theta. 2 2 - sin
.times. .times. .theta. 2 ) .times. r n .times. cos .times. .times.
.PHI. + ( cos .times. .times. .theta. 2 + sin .times. .times.
.theta. 2 ) .times. r n .times. sin .times. .times. .PHI. ( - 1 2 +
cos .times. .times. .theta. 2 2 - sin .times. .times. .theta. 2 )
.times. r n .times. cos .times. .times. .PHI. + ( cos .times.
.times. .theta. 2 - sin .times. .times. .theta. 2 ) .times. r n
.times. sin .times. .times. .PHI. ( - 1 2 + cos .times. .times.
.theta. 2 ) .times. r n .times. cos .times. .times. .PHI. + sin
.times. .times. .theta. 2 .times. r n .times. sin .times. .times.
.PHI. ] ( 14 ) [ x ' y ' z ' ] = [ 1 4 .times. r n .function. ( 2
.times. cos .times. .times. .PHI. + 2 .times. sin .times. .times.
.theta. .function. ( - cos .times. .times. .PHI. + sin .times.
.times. .PHI. ) + 2 .times. cos .times. .times. .theta. .function.
( cos .times. .times. .PHI. + 2 .times. sin .times. .times. .PHI. )
) 1 4 .times. r n .function. ( - 2 .times. cos .times. .times.
.PHI. - 2 .times. cos .times. .times. .theta. .function. ( cos
.times. .times. .PHI. - 2 .times. sin .times. .times. .PHI. ) - 2
.times. sin .times. .times. .theta. .function. ( cos .times.
.times. .PHI. + sin .times. .times. .PHI. ) ) 1 2 .times. r n
.function. ( ( - 1 + cos .times. .times. .theta. ) .times. cos
.times. .times. .PHI. + 2 .times. sin .times. .times. .theta.
.times. .times. sin .times. .times. .PHI. ) ] ( 15 ) ##EQU23##
[0061] The orbitsphere-cvf component of STEP TWO that is generated
by the rotation of a great circle in the xy-plane about the
(-i.sub.x,0i.sub.y,i.sub.z)-axis by 2.pi. followed by a rotation
about the z-axis by .pi./4 corresponding to the output of the
matrix given by Eq. (15) is shown in FIG. 6 wherein the sign of
.phi. is positive for 0.ltoreq..theta..ltoreq.2.pi. in order to
give the angular momentum projections given in the Orbitsphere
Equation of Motion for l=0 section. (The eigenvector ( - 1 2
.times. i x , 1 2 .times. i y , i z ) ##EQU24## of the matrix given
by Eq. (20) corresponding to a rotation about this axis can also be
obtained by phase-shifting .phi. in Eqs. (14-15)) by .pi./4. The
phase shift is given by a shift of time such that .phi. is
phase-shifted by .pi./4. Since the current on each loop is
continuous and periodic, this mathematical aspect does not change
the physical current-density as readily appreciated by comparing
FIG. 6 to FIG. 7. However, phase shifts with respect to .theta. do
have an impact. Phase matching in a convolution operation to
generate the uniform function Y.sub.0.sup.0(.theta.,.phi.) is
discussed in the Matching Phase, Angular Momentum, and Orientation
section.)
[0062] Great Circle in the yz-Plane About the
(-i.sub.x,0i.sub.y,i.sub.z)-Axis by 2,T Followed by a Rotation
About the z-Axis by .pi./4
[0063] Alternatively, STEP TWO comprises the rotation of a great
circle in the yz-plane about the (-i.sub.x,0i.sub.y,i.sub.z)-axis
by 2.pi. followed by a rotation about the z-axis by .pi./4. The
coordinates of the great circle are given by the matrix: [ x ' y '
z ' ] = [ 0 r n .times. cos .times. .times. .PHI. r n .times. sin
.times. .times. .PHI. ] ( 16 ) ##EQU25##
[0064] The matrix for the rotation about the
(-i.sub.x,0i.sub.y,i.sub.z)-axis by 2.pi. followed by a rotation
about the z-axis by .pi./4 is given using Eq. (13) wherein a is
varied from .theta. to 2.pi. and the sign of .phi. is negative for
0.ltoreq..theta..ltoreq.2.pi. in order to give the angular momentum
projections given in the Orbitsphere Equation of Motion for l=0
section. The current pattern that is equivalent to that shown in
FIG. 6 is shown in FIG. 7.
[0065] STEP TWO by Rotation of a Great Circle About the ( - 1 2
.times. i x , 1 2 .times. i y , i z ) ##EQU26## -Axis by 2.pi.
[0066] The orbitsphere-cvf component of STEP TWO is also generated
by the rotation of a great circle about the ( - 1 2 .times. i x , 1
2 .times. i y , i z ) ##EQU27## -axis by 2.pi. wherein the
basis-element great circle bisects the xy-quadrant and is parallel
to the z-axis. The coordinates of the great circle are given by the
matrix that rotates a great circle in the yz-plane about the z-axis
by .pi./4: [ x ' y ' z ' ] = [ cos .times. .times. ( .pi. 4 ) sin
.times. .times. ( .pi. 4 ) 0 - sin .times. .times. ( .pi. 4 ) cos
.times. .times. ( .pi. 4 ) 0 0 0 1 ] .function. [ 0 r n .times. cos
.times. .times. .PHI. r n .times. sin .times. .times. .PHI. ] ( 17
) [ x ' y ' z ' ] = [ sin .times. .times. ( .pi. 4 ) .times. r n
.times. cos .times. .times. .PHI. cos .times. .times. ( .pi. 4 )
.times. r n .times. cos .times. .times. .PHI. r n .times. sin
.times. .times. .PHI. ] = [ r n .times. cos .times. .times. .PHI. 2
r n .times. cos .times. .times. .PHI. 2 r n .times. sin .times.
.times. .PHI. ] ( 18 ) ##EQU28##
[0067] Since STEP TWO is given by the rotation of the yz-plane
basis-element great circle (Eq. (16)) about the
(-i.sub.x,0i.sub.y,i.sub.z)-axis by 2.pi. followed by a rotation
about the z-axis by .pi./4 using Eq. (13), the equivalent result
may be obtained by first rotating the great circle given by Eq.
(18) about the z-axis by -.pi./4, zrot = ( - .pi. 4 ) , ##EQU29##
then applying Eq. (13): [ x ' y ' z ' ] = [ 1 2 + cos .times.
.times. .PHI. 2 2 - sin .times. .times. .theta. 2 cos .times.
.times. .theta. 2 + sin .times. .times. .theta. 2 - 1 2 + cos
.times. .times. .PHI. 2 2 - sin .times. .times. .theta. 2 - 1 2 +
cos .times. .times. .theta. 2 2 - sin .times. .times. .theta. 2 cos
.times. .times. .theta. 2 - sin .times. .times. .theta. 2 - 1 2 +
cos .times. .times. .theta. 2 2 - sin .times. .times. .theta. 2 - 1
2 + cos .times. .times. .theta. 2 sin .times. .times. .theta. 2 1 2
+ cos .times. .times. .theta. 2 ] zrot .function. ( - .pi. 4 ) [ r
n .times. cos .times. .times. .PHI. 2 r n .times. cos .times.
.times. .PHI. 2 r n .times. sin .times. .times. .PHI. ] ( 19 )
##EQU30##
[0068] Using zrot .function. ( - .pi. 4 ) ##EQU31## from Eq. (2)
gives [ x ' y ' z ' ] = [ .times. 1 4 .times. ( 1 + 3 .times.
.times. cos .times. .times. .theta. ) .times. 1 4 .times. ( - 1 +
cos .times. .times. .theta. + 2 .times. 2 .times. sin .times.
.times. .theta. ) .times. 1 4 .times. ( - 2 + 2 .times. cos .times.
.times. .theta. - 2 .times. .times. sin .times. .times. .theta. )
.times. 1 4 .times. ( - 1 + cos .times. .times. .theta. - 2 .times.
2 .times. sin .times. .times. .theta. ) .times. 1 4 .times. ( 1 + 3
.times. .times. cos .times. .times. .theta. ) .times. 1 4 .times. (
2 - 2 .times. cos .times. .times. .theta. - 2 .times. .times. sin
.times. .times. .theta. ) .times. 1 2 .times. ( - 1 + cos .times.
.times. .theta. 2 + sin .times. .times. .theta. ) .times. 1 4
.times. ( 2 - 2 .times. cos .times. .times. .theta. + 2 .times.
.times. sin .times. .times. .theta. ) .times. cos 2 .times. .theta.
2 ] .function. [ r n .times. cos .times. .times. .PHI. 2 r n
.times. cos .times. .times. .PHI. 2 r n .times. sin .times. .times.
.PHI. ] i . .times. ( 20 ) [ x ' y ' z ' ] = [ .times. ( 1 + 3
.times. cos .times. .times. .theta. ) .times. r n .times. cos
.times. .times. .PHI. 4 .times. 2 + ( - 1 + cos .times. .times.
.theta. + 2 .times. 2 .times. sin .times. .times. .theta. ) .times.
r n .times. cos .times. .times. .PHI. 4 .times. 2 + 1 4 .times. ( -
2 + 2 .times. cos .times. .times. .theta. - 2 .times. .times. sin
.times. .times. .theta. ) .times. r n .times. sin .times. .times.
.PHI. .times. ( 1 + 3 .times. cos .times. .times. .theta. ) .times.
r n .times. cos .times. .times. .PHI. 4 .times. 2 + ( - 1 + cos
.times. .times. .theta. - 2 .times. 2 .times. sin .times. .times.
.theta. ) .times. r n .times. cos .times. .times. .PHI. 4 .times. 2
+ 1 4 .times. ( 2 - 2 .times. cos .times. .times. .theta. - 2
.times. .times. sin .times. .times. .theta. ) .times. r n .times.
sin .times. .times. .PHI. .times. ( - 1 + cos .times. .times.
.theta. 2 + sin .times. .times. .theta. ) .times. r n .times. cos
.times. .times. .PHI. 2 .times. 2 + ( 2 - 2 .times. .times. cos
.times. .times. .theta. + 2 .times. .times. sin .times. .times.
.theta. ) .times. r n .times. cos .times. .times. .PHI. 4 .times. 2
+ cos 2 .times. .theta. 2 .times. r n .times. sin .times. .times.
.PHI. ] i . .times. ( 21 ) [ x ' y ' z ' ] = [ .times. 1 4 .times.
r n .function. ( 2 .times. .times. sin .times. .times. .theta.
.function. ( cos .times. .times. .PHI. - sin .times. .times. .PHI.
) + 2 .times. ( 2 .times. .times. cos .times. .times. .theta.
.times. .times. cos .times. .times. .PHI. + ( - 1 + cos .times.
.times. .theta. ) .times. sin .times. .times. .PHI. ) ) .times. 1 4
.times. r n .function. ( - 2 .times. .times. sin .times. .times.
.theta. .function. ( cos .times. .times. .PHI. + sin .times.
.times. .PHI. ) + 2 .times. ( cos .times. .times. .theta. .times.
.times. ( 2 .times. .times. cos .times. .times. .PHI. - sin .times.
.times. .PHI. ) + sin .times. .times. .PHI. ) ) .times. 1 2 .times.
r n .function. ( 2 .times. cos .times. .times. .PHI. .times.
.times. sin .times. .times. .theta. + ( 1 + cos .times. .times.
.theta. ) .times. .times. sin .times. .times. .PHI. ) ] i . .times.
( 22 ) ##EQU32##
[0069] where the sign of .phi. is negative for
0.ltoreq..theta..ltoreq.2.pi. in order to give the angular momentum
projections given in the Orbitsphere Equation of Motion for l=0
section of Mills GUT. The current pattern of the orbitsphere-cvf
component of STEP TWO that is generated by the rotation of a great
circle in the xyz-plane about the ( - 1 2 .times. i x , 1 2 .times.
i y , i z ) ##EQU33## -axis by 2.pi. corresponding to the output of
the matrices given by Eqs. (20-22) is equivalent to that shown in
FIGS. 6 and 7. In the ( - 1 2 .times. i x , 1 2 .times. i y , i z )
##EQU34## -rotational-axis case, the great circle current loop that
serves as a basis element and initially and finally bisects the
xy-quadrant and is parallel to the z-axis is shown as red in FIG.
7.
[0070] The current pattern of the orbitsphere-cvf component of STEP
TWO shown in FIG. 6 is also generated by the rotation of a great
circle in the xy-plane about the ( - 1 2 .times. i x , 1 2 .times.
i y , i z ) ##EQU35## -axis by 2.pi.. Here, the great circle given
by [ x ' y ' z ' ] = [ r n .times. cos .times. .times. .PHI. r n
.times. sin .times. .times. .PHI. 0 ] ( 23 ) ##EQU36##
[0071] is input to transformational matrix given by Eq. (20), and
the sign of .phi. is positive for 0.ltoreq..theta..ltoreq.2.pi. in
order to give the angular momentum projections given in the
Orbitsphere Equation of Motion for l=0 section of Mills GUT.
[0072] Characteristics of the Orbitsphere-cvf
[0073] From Eqs. (1.76) and (1.77), angular momentum components of
the orbitsphere-cvf are L xy = 4 .times. .times. and .times.
.times. L z = 2 . ##EQU37## The corresponding resultant angular
momentum vector, L.sub.R, has magnitude {square root over (5)}h/4
along the direction of the spherical-coordinate angles
.theta.=0.4636 rad, .PHI. = 3 .times. .pi. 4 .times. rad .
##EQU38## To obtain the view along L.sub.R, the orbitsphere-cvf is
first rotated counterclockwise about the vector
(i.sub.x,i.sub.y,0i.sub.z) by an angle -0.4636 rad using Eq. (4) or
Eq. (1.69) wherein .DELTA..alpha..sub.x' and .DELTA..alpha..sub.y'.
are each - {square root over (2)}(0.4636) rad to align L.sub.R with
the z-axis as shown in FIG. 8.
[0074] The angular momentum is constant with respect to rotation of
the orbitsphere-cvf about the axis of the resultant angular
momentum vector, L.sub.R. In this case, the corresponding component
angular momentum L.sub.xy is rotationally constant about the
xy-axis, and the corresponding L.sub.x and L.sub.y components are
rotationally constant about the x- and y-axes, respectively. The
component L.sub.z is further rotationally constant about the
z-axis. The constancy of the angular momentum with respect to
rotation of the orbitsphere-cvf about each of the principal axes
determines that the corresponding rotational symmetry of each axis
is C.sub..infin. for the corresponding component even though the
spatial symmetry of the current distribution is less. The
angular-momentum axis of each component orbitsphere-cvf
corresponding to either STEP ONE or STEP TWO is also a
C.sub..infin.-axis. Each component is comprised of great circles,
and each great circle has a spatial and angular-momentum
C.sub..infin.-axis perpendicular to the plane it defines. In
addition, each component as well as the orbitsphere-cvf has the
origin as a spatial inversion center (C.sub.i) as shown in FIGS.
1.10A-F of Mills GUT.
[0075] Each component orbitsphere-cvf has an infinite number of
spatial C.sub.2-axes that lie in a symmetry plane
(.sigma..sub..nu.-plane) with a perpendicular spatial
C.sub..infin.-axis. Consider that the C.sub.2- and
C.sub..infin.-axes shown in FIG. 9 are defined as the z-axis and
the y-axis of the STEP-ONE component, respectively. The x-axis is
perpendicular to the y- and z-axes. Then, the
.sigma..sub..nu.-plane is the xz-plane. The parameters of the width
a from the center at the position of the .sigma..sub..nu.-plane to
the edge of the STEP-ONE component of the orbitsphere-cvf and the
distance b from the edge to the apex of a circle defined by the
orthogonal STEP-TWO component orbitsphere-cvf are shown in FIG. 9.
Also shown is the angle .theta. between the C.sub..infin.-axis
(y-axis) of the STEP-ONE component and the intersection of the edge
of the two orthogonal component orbitsphere-cvfs. By symmetry, this
is also the angle between the .sigma..sub..nu.-plane and the edge.
These parameters can be related to the radius r.sub.n of the
orbitsphere-cvf. From FIG. 9, the relationship between the radius
and the width is a.sup.2+a.sup.2=r.sub.n.sup.2 (24)
[0076] Thus, the width is a = r n 2 .times. .times. Since ( 25 ) a
+ b = r n ( 26 ) ##EQU39##
[0077] the distance from the edge of the STEP-ONE component to the
to the apex at the STEP-TWO component using Eq. (25) is b = r n
.function. ( 1 - 1 2 ) ( 27 ) ##EQU40##
[0078] The angle .theta. is then .theta. = cos - 1 .times. r n 2 r
n = .pi. 4 ( 28 ) ##EQU41##
[0079] As shown in FIG. 9, the current density of the STEP-ONE
component orbitsphere-cvf is constant along the C.sub.2-axis
(z-axis) and increases from the origin to the edge along the
C.sub..infin.-axis. Since the current is on a sphere, the
corresponding polar coordinates along this span are from .PHI. = 0
.times. .times. to .times. .times. .PHI. = .pi. 4 . ##EQU42## It
can be appreciated from FIG. 2 that at .phi.=0, the great circles
are initially at an angle of .theta. ' = .pi. 4 ##EQU43## relative
to the C.sub.2-axis as defined in FIG. 9 and are at an angle
.theta. ' = 0 .times. .times. at .times. .times. .PHI. = .pi. 4 .
##EQU44## Thus, the differential area per loop is given by the
cosine of the angle .PHI. + .pi. 4 , ##EQU45## and the density D is
given by the inverse of the area function: D = sec .function. (
.PHI. + .pi. 4 ) ( 29 ) ##EQU46##
[0080] The Uniform Current (Charge)-Density Function
Y.sub.0.sup.0(.theta.,.phi.)
[0081] Boundary Constraints
[0082] The further constraint that the current density is uniform
such that the charge density is uniform, corresponding to an
equipotential, minimum energy surface is exactly satisfied by using
the orbitsphere-cvf as a basis element to generate
Y.sub.0.sup.0(.theta.,.phi.). Utilizing the symmetry properties of
each component of the orbitsphere-cvf corresponding to either STEP
ONE or STEP TWO and the orthonormality of the trigonometric
functions that generate the orbitsphere-cvf, a convolution operator
comprising an autocorrelation-type function [2] gives rise to the
spherically-symmetric current density,
Y.sub.0.sup.0(.theta.,.phi.). The operator comprises the
convolution of each great circle current loop of the
orbitsphere-cvf designated as the primary orbitsphere-cvf with a
second orbitsphere-cvf designated as the secondary orbitsphere-cvf.
The angular momenta of the convolved elements are matched. The
elements are also orientation matched by rotation of the secondary
about the appropriate axis (axes), and the elements are phase
matched using a rotation of each secondary orbitsphere-cvf element
about its C.sub..infin.-axis. The convolution is over the angular
span .theta.=0 to .theta.=2.pi. corresponding to the rotation of
the basis-current loop which generated the primary orbitsphere-cvf.
The angular momenta of the secondary elements project onto the
resultant angular momentum axis, L.sub.R-axis, of the primary
orbitsphere- cvf equivalently to those of its great circles. The
resulting exact uniform current distribution obtained from the
convolution has the same angular momentum distribution, resultant,
L.sub.R, and components of L xy = 4 .times. .times. and .times.
.times. L z = 2 ##EQU47## as those of the orbitsphere-cvf used as a
primary basis element.
[0083] Properties of the Orbitsphere-cvf Permissive to Generate
Y.sub.0.sup.0(.theta.,.phi.)
[0084] First, consider the symmetry properties of the each of the
two orthogonal components the orbitsphere-cvf corresponding to STEP
ONE and STEP TWO. As shown in the previous sections and the
Orbitsphere Equation of Motion for l=0 section, a basis-element
great circle is rotated by 2.pi. to generate the same current
pattern as that generated over the surface by rotation of two
orthogonal great circle current loops by .pi. using Eqs. (1.69) and
(1.71). The resulting component orbitsphere-cvf is always
perpendicular to the 2.pi.-axis of rotation used to generate the
component from the great circle. This 2.pi.-rotational axis defines
a unique C.sub..infin.-axis which serves as a vector to
characterize the angular momentum of the orbitsphere-cvf great
circles. The resultant angular momentum of any given nth pair of
great-circle elements defined by the rotational angle of the nth
great circle at .theta.=.theta..sub.n and its orthogonal partner at
.theta.=.theta..sub.n+.pi. may be along the C.sub..infin.-axis or
perpendicular to it depending on the current direction of the great
circles for 0.ltoreq..theta..ltoreq..pi. and
.pi..ltoreq..THETA..ltoreq.2.pi.. In the case that the angular
momentum is perpendicular to the C.sub..infin.-axis, the angular
momentum vector rotates about the C.sub..infin.-axis by .pi. as a
function of .theta. of Eq. (5) or as a function of .theta. of Eqs.
(1.72-1.73) as shown in FIG. 1.11 of Mills GUT. Also, shown in FIG.
1.11 is an example of a stationary angular momentum vector that
results when the angular momentum of each pair of great circles is
along the C.sub..infin.-axis.
[0085] Component Orbitsphere-cvf Orthogonal to that of STEP ONE be
the Rotation of a Great Circle About the
(i.sub.x,-i.sub.y,0i.sub.z)-Axis by 2.pi.
[0086] As a further example of a stationary orbitsphere-cvf angular
momentum vector, consider the case of STEP ONE using the great
circle current loops shown in FIG. 1.4 of Mills GUT as the basis
elements. The resultant angular momentum vector of magnitude 2
.times. 2 ##EQU48##
[0087] shown in FIG. 1.4 is stationary throughout the rotations of
the orthogonal-great-circle basis set about the
(i.sub.x,-i.sub.y,0i.sub.z)-axis by an angle .pi. or one of the
great circles by 2.pi. that transform the axes as given in Table 1
during the generation of the component of the orbitsphere-cvf. As
shown by comparing FIGS. 2 and 11, the resulting component
orbitsphere-cvf called STEP ONE .perp. is orthogonal to that of
STEP ONE given by rotation of the basis-element(s) great circle(s)
about the perpendicular axis of (i.sub.x,-i.sub.y,0i.sub.z).
TABLE-US-00001 TALE 1 Summary of the results of the rotation of the
two orthogonal current loops in the xz- and yz-planes about the
(i.sub.x, -i.sub.y, 0i.sub.z) -axis to generate a component
orbitsphere-cvf with a stationary angular momentum vector. Step
Initial Direction Final Direction Initial to Final L.sub.x L.sub.y
of Angular of Angular Axis Momentum Momentum Transformation
Components Components ({circumflex over (r)} .times. {circumflex
over (K)}).sup.a ({circumflex over (r)} .times. {circumflex over
(K)}).sup.a 1 .perp. ##EQU49## {circumflex over (x)}, -y
{circumflex over (x)}, -y +x' .fwdarw. -y 4 ##EQU50## - 4 ##EQU51##
+y' .fwdarw. -x +z' .fwdarw. -z a .times. K .times. .times. is
.times. .times. the .times. .times. current .times. .times. density
, r .times. .times. is .times. .times. the .times. .times. polar
.times. .times. vector .times. .times. of .times. .times. the
.times. .times. great .times. .times. circle , and .times. " ^ "
.times. .times. denotes .times. .times. the .times. .times. unit
.times. .times. vectors .times. .times. u ^ .ident. u u .
##EQU52##
[0088] Great Circle in the yz-Plane About the
(i.sub.x,-i.sub.y,0i.sub.z)-Axis
[0089] Following the procedure given in Fowles [1], the component
orbitsphere-cvf that is orthogonal to that of STEP ONE (STEP ONE
.perp.) is generated by the rotation of a great circle in the
yz-plane about the (i.sub.x,-i.sub.y,0i.sub.z)-axis by 2.pi.. A
first transformation matrix is generated by the combined rotation
of a great circle in the yz-plane about the z-axis by - .pi. 4
##EQU53## then about the x-axis by .theta. where positive rotations
about an axis are defined as clockwise: [ x ' y ' z ' ] = [ cos
.function. ( .pi. 4 ) - sin .function. ( .pi. 4 ) 0 sin .function.
( .pi. 4 ) .times. cos .times. .times. .theta. cos .function. (
.pi. 4 ) .times. cos .times. .times. .theta. sin .times. .times.
.theta. - sin .function. ( .pi. 4 ) .times. sin .times. .times.
.theta. - cos .function. ( .pi. 4 ) .times. sin .times. .times.
.theta. cos .times. .times. .theta. ] .function. [ 0 r n .times.
cos .times. .times. .PHI. r n .times. sin .times. .times. .PHI. ] (
30 ) ##EQU54##
[0090] The transformation matrix about (i.sub.x,-i.sub.y,0i.sub.z)
is given by multiplication of the output of the matrix given by Eq.
(30) by the matrix corresponding to a rotation about the z-axis of
.pi. 4 . ##EQU55## The output of the matrix given by Eq. (30) is
shown in FIG. 10 wherein .theta. is varied from 0 to 2.pi..
[0091] The rotation matrix about the z-axis by .pi./4, zrot
.function. ( .pi. 4 ) , ##EQU56## is given by Eq. (1 1). Thus, [ x
' y ' z ' ] = zrot .function. ( .pi. 4 ) [ cos .function. ( .pi. 4
) - sin .function. ( .pi. 4 ) 0 sin .function. ( .pi. 4 ) .times.
cos .times. .times. .theta. cos .function. ( .pi. 4 ) .times. cos
.times. .times. .theta. sin .times. .times. .theta. - sin
.function. ( .pi. 4 ) .times. sin .times. .times. .theta. - cos
.function. ( .pi. 4 ) .times. sin .times. .times. .theta. cos
.times. .times. .theta. ] .function. [ 0 r n .times. cos .times.
.times. .PHI. r n .times. sin .times. .times. .PHI. ] ( 31 )
##EQU57##
[0092] Substitution of the matrix given by Eq. (11) into Eq. (31)
gives [ x ' y ' z ' ] = [ 1 2 + cos .times. .times. .theta. 2 - 1 2
+ cos .times. .times. .theta. 2 sin .times. .times. .theta. 2 - 1 2
+ cos .times. .times. .theta. 2 1 2 + cos .times. .times. .theta. 2
sin .times. .times. .theta. 2 - sin .times. .times. .theta. 2 - sin
.times. .times. .theta. 2 cos .times. .times. .theta. ] .function.
[ 0 r n .times. cos .times. .times. .PHI. r n .times. sin .times.
.times. .PHI. ] ( 32 ) [ x ' y ' z ' ] = [ ( - 1 2 + cos .times.
.times. .theta. 2 ) .times. r n .times. cos .times. .times. .PHI. +
sin .times. .times. .theta. 2 .times. r n .times. sin .times.
.times. .PHI. ( 1 2 + cos .times. .times. .theta. 2 ) .times. r n
.times. cos .times. .times. .PHI. + sin .times. .times. .theta. 2
.times. r n .times. sin .times. .times. .PHI. - sin .times. .times.
.theta. 2 .times. r n .times. cos .times. .times. .PHI. + cos
.times. .times. .theta. .times. .times. r n .times. sin .times.
.times. .PHI. ] ( 33 ) ##EQU58##
[0093] The component orbitsphere-cvf that is orthogonal to that of
STEP ONE that is generated by the rotation of a great circle in the
yz-plane about the (i.sub.x,-i.sub.y,0i.sub.z)-axis by 2.pi.
corresponding to the output of the matrix given by Eq. (33) is
shown in FIG. 11 wherein the sign of .phi. is positive for
0.ltoreq..theta..ltoreq..pi. and negative for
.pi..ltoreq..theta..ltoreq.2.pi. in order to give the currents
directions shown in FIG. 1.4 of Mills GUT. The angular momentum
vector is stationary along the (i.sub.x,-i.sub.y,0i.sub.z)-axis as
shown in FIG. 1.4 of the Orbitsphere Equation of Motion for l=0
section of Mills GUT. The same result is given by the rotation of a
great circle in the xz-plane about the
(i.sub.x,-i.sub.y,0i.sub.z)-axis by 2.pi. using Eqs. (6) and
(32).
[0094] Matching Phase, Angular Momentum, and Orientation
[0095] For STEP ONE .perp., the resultant angular momentum vector,
L.sub.R, is along (i.sub.x,-i.sub.y,0i.sub.z). The angular momentum
is constant for any rotation about the axis; thus, it is a
C.sub..infin.-axis relative to the angular momentum. However,
rotation about this axis does change the phase (coordinate position
relative to the starting position) of the component
orbitsphere-cvf. For example, a rotation by .pi. about the
(i.sub.x,-i.sub.y,0i.sub.z)-axis using Eq. (32) causes the
basis-element great circle to rotate by .pi. 2 ##EQU59## about the
z-axis as shown in FIG. 12. The rotation matrix about the
(i.sub.x,-i.sub.y,0i.sub.z)-axis by .pi.,
(i.sub.x,-i.sub.y,0i.sub.z)rot(.pi.), is given by ( i x , - i y , 0
.times. i z ) .times. rot .function. ( .pi. ) = [ 1 2 + cos .times.
.times. .pi. 2 - 1 2 + cos .times. .times. .pi. 2 sin .times.
.times. .pi. 2 - 1 2 + cos .times. .times. .pi. 2 1 2 + cos .times.
.times. .pi. 2 sin .times. .times. .pi. 2 - sin .times. .times.
.pi. 2 - sin .times. .times. .pi. 2 cos .times. .times. .pi. ] = [
0 - 1 0 - 1 0 0 0 0 - 1 ] .times. .times. Thus , ( 34 ) [ x ' y ' z
' ] = ( i x , - i y , 0 .times. i z ) .times. rot .function. ( .pi.
) [ 1 2 + cos .times. .times. .theta. 2 - 1 2 + cos .times. .times.
.theta. 2 sin .times. .times. .theta. 2 - 1 2 + cos .times. .times.
.theta. 2 1 2 + cos .times. .times. .theta. 2 sin .times. .times.
.theta. 2 - sin .times. .times. .theta. 2 - sin .times. .times.
.theta. 2 cos .times. .times. .theta. ] .function. [ 0 r n .times.
cos .times. .times. .PHI. r n .times. sin .times. .times. .PHI. ] (
35 ) ##EQU60##
[0096] Substitution of the matrix given by Eq. (34) into Eq. (35)
gives the current pattern for the .pi. 2 ##EQU61## phase shift
relative to the z-axis corresponding to a rotation of the component
orbitsphere-cvf given by Eq. (32) about the
(i.sub.x,-i.sub.y,0i.sub.z)-axis by .pi. as shown in FIG. 12: [ x '
y ' z ' ] = [ sin 2 .times. .theta. 2 - cos 2 .times. .theta. 2 -
sin .times. .times. .theta. 2 - cos 2 .times. .theta. 2 sin 2
.times. .theta. 2 - sin .times. .times. .theta. 2 sin .times.
.times. .theta. 2 sin .times. .times. .theta. 2 - cos .times.
.times. .theta. ] .function. [ 0 r n .times. cos .times. .times.
.PHI. r n .times. sin .times. .times. .PHI. ] ( 36 ) [ x ' y ' z '
] = [ - cos 2 .times. .theta. 2 .times. r n .times. cos .times.
.times. .PHI. - sin .times. .times. .theta. 2 .times. r n .times.
sin .times. .times. .PHI. sin 2 .times. .theta. 2 .times. r n
.times. cos .times. .times. .PHI. - sin .times. .times. .theta. 2
.times. r n .times. sin .times. .times. .PHI. sin .times. .times.
.theta. 2 .times. r n .times. cos .times. .times. .PHI. - cos
.times. .times. .theta. .times. .times. r n .times. sin .times.
.times. .PHI. ] ( 37 ) ##EQU62##
[0097] In general, for the stationary-angular-momentum-vector case,
the angular momentum vector along the C.sub..infin.-axis is always
at an angle of .pi. 4 ##EQU63## relative to the plane of the
basis-element great circle that generated the component
orbitsphere-cvf. Thus, the plane of the great circle relative to
the xyz coordinate system can be rotated over a span of .+-. .pi. 4
##EQU64## rotation about L.sub.R. The rotation of the basis-element
great circle corresponds to changing the phase of the component
orbitsphere-cvf in Eq. (38). The phase of the secondary component
orbitsphere-cvf can be matched to that of the basis-element great
circle of a primary component orbitsphere-cvf by rotation about
L.sub.R. Furthermore, each component orbitsphere-cvf is comprised
of great circles, and each great circle has a C.sub..infin.-axis
perpendicular to the plane. This feature further permits the phase
within the great circles of the secondary component orbitsphere-cvf
to be matched to that of the basis element great circle of the
primary.
[0098] Since the angular momentum vector is stationary and is a
C.sub..infin.-axis, the secondary component orbitsphere-cvf can be
made to match the angular momentum of the basis-element great
circle of any primary component orbitsphere-cvf by rotations that
align the vector of the former with that of the latter. In addition
to phase and angular momentum, the orientation of the secondary
component orbitsphere-cvf is matched to that of the great circles
elements of the primary, by rotation about the appropriate axis
that aligns the angular momenta and orientation of the secondary
component orbitsphere-cvf with the basis-element great circle of
the primary. Thus, the secondary component orbitsphere-cvf serves
as a basis element in a convolution operator as shown in the
Convolution Operator section.
[0099] Convolution Operator
[0100] The orbitsphere-cvf comprises two components corresponding
to each of STEP ONE and STEP TWO. A uniform current-density
function can be generated from each component independently having
the same resultant angular momentum as the corresponding component,
and the two uniform functions can be superimposed to give
Y.sub.0.sup.0(.theta.,.phi.).
[0101] Consider the component STEP ONE .perp. generated in the
Component Orbitsphere-cvf Orthogonal to that of STEP ONE by
Rotation of a Great Circle about the
(i.sub.x,-i.sub.y,0i.sub.z)-axis by 2.pi. section. The component is
generated by rotation of the two orthogonal great circles about the
(i.sub.x,-i.sub.y,0i.sub.z)-axis by .pi. or the rotational of one
of the loops about this axis by 2.pi.. In the former case, the
resultant angular momenta of the pair of orthogonal great circle
current loops and that of the component orbitsphere-cvf are along
this rotational axis, and in the latter case, the resultant angular
momentum of the component orbitsphere-cvf is equivalent to that of
the former.
[0102] When the resultant angular of the component orbitsphere-cvf
is along the axis about which the basis elements or element are
rotated to generate the component orbitsphere-cvf, the angular
momentum of the basis elements or elements in conserved in the
superposition of the rotated basis elements or element. In the case
that one basis element is used, the rotational axis is the
2.pi.-rotational axis of the Properties of the Orbitsphere-cvf
Permissive to Generate Y.sub.0.sup.0(.theta.,.phi.) section. Thus,
in the general case that the resultant angular momentum of the
component orbitsphere-cvf is along the 2.pi.-rotational axis, a
secondary nth component orbitsphere-cvf can serve as a basis
element to match the angular momentum of any given nth great circle
of a primary component orbitsphere-cvf. The replacement of each
great circle of the primary orbitsphere-cvf with a secondary
orbitsphere-cvf of matching angular momentum, orientation, and
phase comprises an autocorrelation-type function [2] that exactly
gives rise to a uniform current density having the same resultant
angular momentum as the corresponding primary component. The
uniform distributions for STEP ONE and STEP TWO can be generated
independently and superimposed to give
Y.sub.0.sup.0(.theta.,.phi.).
[0103] The orbitsphere-cvf shown in FIGS. 1.10A-G of Mills GUT and
8 and 9 comprises the superposition or sum of the components
corresponding to STEPS ONE (FIGS. 2 and 3) and STEP TWO (FIGS. 6
and 7). Thus, the convolution is performed on each component
designated a primary component. The convolution of a secondary
component orbitsphere-cvf element with the each great circle
current loop of each primary orbitsphere-cvf is designated as the
convolution operator, Y.sub.0.sup.0(.theta.,.phi.), given by A
.function. ( .theta. , .PHI. ) = .times. 1 2 .times. r n 2 .times.
lim .DELTA..theta. 2 .fwdarw. 0 .times. m ' = 1 m ' = 2 .times.
.pi. .DELTA..theta. 2 .times. lim .DELTA..theta. 1 .fwdarw. 0
.times. m = 1 m = 2 .times. .pi. .DELTA..theta. 1 .times. 2 .times.
.degree.O .function. ( .theta. , .PHI. ) .times. ( 1 1 .degree.
.times. 0 .times. ( .theta. , .PHI. ) .times. .delta. .function. (
.theta. - m .times. .times. .DELTA..theta. 1 , .PHI. - .PHI. ' ) +
1 2 .degree. .times. 0 .times. ( .theta. , .PHI. ) .times. .delta.
.function. ( .theta. - m ' .times. .DELTA..theta. 2 , .PHI. - .PHI.
'' ) ) = .times. 1 2 .times. r n 2 .times. lim .DELTA..theta. 2
.fwdarw. 0 .times. m ' = 1 m ' = 2 .times. .pi. .DELTA..theta. 2
.times. lim .DELTA..theta. 1 .fwdarw. 0 .times. m = 1 m = 2 .times.
.pi. .DELTA..theta. 1 .times. 2 .times. .degree.O .function. (
.theta. , .PHI. ) .times. ( GC STEP .times. .times. ONE .function.
( m .times. .times. .DELTA..theta. 1 , .PHI. 1 ) + GC STEP .times.
.times. TWO .function. ( m ' .times. .DELTA..theta. 2 , .PHI. '' )
) ( 38 ) ##EQU65##
[0104] wherein (1) the secondary component orbitsphere-cvf that is
matched to the basis element of the primary is defined by the
symbol 2.degree.O(.theta.,.phi.), (2) the primary component
orbitsphere-cvf of STEP M is defined by the symbol
1.degree..sub.MO(.theta.,.phi.), (3) each rotated great circle of
the primary component orbitsphere-cvf of STEP M is selected by the
Dirac delta function
.delta.(.theta.-m.DELTA..theta..sub.M,.phi.-.phi.',); the product
1.degree..sub.MO(.theta.,.phi.).delta.(.theta.-m.DELTA..sub.M,.phi.-.phi.-
') is zero except for the great circle at the angle
.theta.=m.DELTA..theta..sub.M about the 2.pi.-rotational axis; each
selected great circle having 0.ltoreq..phi.'.ltoreq.2.pi. is
defined by GC.sub.STEPM (m.DELTA..theta..sub.M,.phi.'), and ( 4 )
.times. 1 2 .times. r n 2 ##EQU66## is the normalization constant.
In Eq. (38), the angular momentum of each secondary component
orbitsphere-cvf is equal in magnitude and direction as that of the
current loop with which it is convolved. Furthermore, the
orientations and phases of the convolved elements are matched by
rotating the secondary component orbitsphere-cvf about the
appropriate principle axis (axes) and about the C.sub..infin.-axis
along its angular momentum vector, respectively. With the magnitude
of the angular momentum of the secondary component orbitsphere-cvf
matching that of the current loop which it replaces during the
convolution and the loop then serving as a unit vector, the angular
momentum resulting from the convolution operation is inherently
normalized to that of the primary component orbitsphere-cvf.
[0105] The convolution of a sum is the sum of the convolutions.
Thus, the convolution operation may be performed on each of STEP
ONE and STEP TWO separately, and the result may be superposed in
terms of the current densities and angular momenta. A .function. (
.theta. , .PHI. ) = 1 2 .times. r n 2 .times. ( lim .DELTA..theta.
2 .fwdarw. 0 .times. m ' = 1 m ' = 2 .times. .pi. .DELTA..theta. 2
.times. 2 .times. .degree.O .function. ( .theta. , .PHI. ) GC STEP
.times. .times. ONE .function. ( m .times. .times. .DELTA..theta. 1
, .PHI. 1 ) lim .DELTA..theta. 1 .fwdarw. 0 .times. m = 1 m = 2
.times. .pi. .DELTA..theta. 1 .times. 2 .times. .degree.O
.function. ( .theta. , .PHI. ) GC STEP .times. .times. TWO
.function. ( m ' .times. .DELTA..theta. 2 , .PHI. '' ) ) ( 39 )
##EQU67##
[0106] Factoring out the secondary component orbitsphere-cvf which
is a constant at each position of GC.sub.STEPM
(m.DELTA..theta..sub.M,.phi.') gives A .function. ( .PHI. , .theta.
) = 1 2 .times. r n 2 .times. 2 .times. .degree.O .function. (
.theta. , .PHI. ) .times. ( lim .DELTA..theta. 2 .fwdarw. 0 .times.
m ' = 1 m ' = 2 .times. .pi. .DELTA..theta. 2 .times. GC STEPONE
.function. ( m .times. .times. .DELTA..theta. 1 , .PHI. 1 ) + lim
.DELTA..theta. 1 .fwdarw. 0 .times. m = 1 m = 2 .times. .pi.
.DELTA..theta. 1 .times. GC STEP .times. .times. TWO .function. ( m
' .times. .DELTA..theta. 2 , .PHI. '' ) ) ( 40 ) ##EQU68##
[0107] The summation is the operator that generates the primary
component orbitsphere-cvf of STEP M,
1.degree..sub.MO(.theta.,.phi.). Thus, the current-density function
is given by the dot product of each primary orbitsphere-cvf with
itself. The result is the scalar sum of the square of each of the
STEP ONE and STEP TWO primary component orbitsphere-cvfs: A
.function. ( .theta. , .PHI. ) = 1 2 .times. r n 2 .times. ( ( 1 1
.degree. .times. O .function. ( .theta. , .PHI. ) ) 2 + ( 1 2
.degree. .times. O .function. ( .theta. , .PHI. ) ) 2 ) ( 41 )
##EQU69##
[0108] where the dot-product scalar is valid over the entire
spherical surface.
[0109] Component Orbitsphere-cvf Squared for STEP ONE Using the
Rotation of a Great Circle About the
(i.sub.x,i.sub.y,0i.sub.z)-Axis by 2.pi.
[0110] The convolution of the secondary orbitsphere-cvf with the
primary orbitsphere-cvf gives the dot product of the primary
orbitsphere-cvf with itself over the entire spherical surface. From
Eq. (5), the corresponding scalar equation for the component
orbitsphere-cvf squared for STEP ONE is given by x 2 + y 2 + z 2 =
[ ( ( 1 2 - cos .times. .times. .theta. 2 ) .times. r n .times. cos
.times. .times. .PHI. - sin .times. .times. .theta. 2 .times. r n
.times. sin .times. .times. .PHI. ) 2 + ( ( 1 2 + cos .times.
.times. .theta. 2 ) .times. r n .times. cos .times. .times. .PHI. +
sin .times. .times. .theta. 2 .times. r n .times. sin .times.
.times. .PHI. ) 2 + ( - sin .times. .times. .theta. 2 .times. r n
.times. cos .times. .times. .PHI. + cos .times. .times. .theta.
.times. .times. r n .times. sin .times. .times. .PHI. ) 2 ] ( 42 )
##EQU70##
[0111] Multiplying out the squared terms gives x 2 + y 2 + z 2 = [
( 1 2 - cos .times. .times. .theta. 2 ) 1 .times. r n 2 .times. cos
2 .times. .times. .PHI. + sin 2 .times. .times. .theta. 2 .times. r
n 2 .times. sin 2 .times. .times. .PHI. - 2 .times. ( 1 2 - cos
.times. .times. .theta. 2 ) .times. r n .times. cos .times. .times.
.PHI. - sin .times. .times. .theta. 2 .times. r n .times. sin
.times. .times. .PHI. + ( 1 2 + cos .times. .times. .theta. 2 ) 2
.times. r n 2 .times. cos 2 .times. .times. .PHI. + sin 2 .times.
.times. .theta. 2 .times. r n 2 .times. sin 2 .times. .times. .PHI.
+ 2 .times. ( 1 2 + cos .times. .times. .theta. 2 ) .times. r n
.times. cos .times. .times. .PHI. .times. sin .times. .times.
.theta. 2 .times. r n .times. sin .times. .times. .PHI. + sin 2
.times. .times. .theta. 2 .times. r n 2 .times. cos 2 .times.
.times. .PHI. + cos 2 .times. .times. .theta. .times. .times. r n 2
.times. sin 2 .times. .times. .PHI. - 2 .times. sin .times. .times.
.theta. 2 .times. r n .times. cos .times. .times. .PHI.cos .times.
.times. .theta. .times. .times. r n .times. sin .times. .times.
.PHI. ] ( 43 ) ##EQU71##
[0112] Further multiplying out the squared terms gives x 2 + y 2 +
z 2 = r n 2 .function. [ ( 1 4 - cos .times. .times. .theta. 2 +
cos 2 .times. .theta. 4 ) .times. cos 2 .times. .PHI. + sin 2
.times. .theta. 2 .times. sin 2 .times. .PHI. - 2 .times. ( 1 2 -
cos .times. .times. .theta. 2 ) .times. cos .times. .times. .PHI.
.times. sin .times. .times. .theta. 2 .times. sin .times. .times.
.PHI. + ( 1 4 + cos .times. .times. .theta. 2 + cos 2 .times.
.theta. 4 ) .times. cos 2 .times. .PHI. + sin 2 .times. .theta. 2
.times. sin 2 .times. .PHI. + 2 .times. ( 1 2 + cos .times. .times.
.theta. 2 ) .times. cos .times. .times. .PHI. .times. sin .times.
.times. .PHI. 2 .times. sin .times. .times. .PHI. + sin 2 .times.
.theta. 2 .times. cos 2 .times. .PHI. + cos 2 .times. .theta.
.times. .times. sin 2 .times. .PHI. - 2 .times. sin .times. .times.
.theta. 2 .times. cos .times. .times. .PHI. .times. .times. cos
.times. .times. .theta.sin .times. .times. .PHI. .times. ] ( 44 )
##EQU72##
[0113] Combining terms gives x 2 + y 2 + z 2 = r n 2 .function. [ (
1 2 + cos 2 .times. .theta. 2 + sin 2 .times. .theta. 2 ) .times.
cos 2 .times. .PHI. + ( sin 2 .times. .theta. + cos 2 .times.
.theta. ) .times. sin 2 .times. .PHI. + 4 .times. ( cos .times.
.times. .theta. 2 ) .times. cos .times. .times. .PHI. .times. sin
.times. .times. .theta. 2 .times. sin .times. .times. .PHI. - 2
.times. sin .times. .times. .theta. 2 .times. cos .times. .times.
.PHI. .times. .times. cos .times. .times. .theta. .times. .times.
sin .times. .times. .PHI. ] ( 45 ) ##EQU73##
[0114] Using the trigonometric identity: sin 2 .times. .theta. +
cos 2 .times. .theta. = 1 .times. .times. gives ( 46 ) x 2 + y 2 +
z 2 = r n 2 .function. [ ( 1 2 + 1 2 ) .times. cos 2 .times. .PHI.
+ sin 2 .times. .PHI. ] ( 47 ) ##EQU74##
[0115] By using the trigonometric identity of Eq. (46) for .phi.,
Eq. (47) becomes x.sup.2+y.sup.2+z.sup.2=r.sub.n.sup.2 (48)
[0116] which is the equation of a uniform sphere.
[0117] Component Orbitsphere-cvf Squared for STEP TWO Using the
Rotation of a Great Circle About the
(-i.sub.x,0i.sub.y,i.sub.z)-Axis by 2.pi. Followed by a Rotation
About the z-Axis by .pi./4
[0118] From Eq. (14), the equation for the component
orbitsphere-cvf squared for STEP TWO is given by x 2 + y 2 + z 2 =
{ ( ( 1 2 + cos .times. .times. .theta. 2 2 - sin .times. .times.
.theta. 2 ) .times. r n .times. cos .times. .times. .PHI. + ( cos
.times. .times. .theta. 2 + sin .times. .times. .theta. 2 ) .times.
r n .times. sin .times. .times. .PHI. ) 2 + ( ( - 1 2 + cos .times.
.times. .theta. 2 2 - sin .times. .times. .theta. 2 ) .times. r n
.times. cos .times. .times. .PHI. + ( cos .times. .times. .theta. 2
- sin .times. .times. .theta. 2 ) .times. r n .times. sin .times.
.times. .PHI. ) 2 + ( ( - 1 2 + cos .times. .times. .theta. 2 )
.times. r n .times. cos .times. .times. .PHI. + sin .times. .times.
.theta. 2 .times. r n .times. sin .times. .times. .PHI. ) 2 } ( 49
) ##EQU75##
[0119] Multiplying out the squared terms gives x 2 + y 2 + z 2 = {
( 1 2 + cos .times. .times. .theta. 2 2 - sin .times. .times.
.theta. 2 ) 2 .times. r n 2 .times. cos 2 .times. .PHI. + ( cos
.times. .times. .theta. 2 + sin .times. .times. .theta. 2 ) 2
.times. r n 2 .times. sin 2 .times. .PHI. + 2 .times. ( 1 2 + cos
.times. .times. .theta. 2 2 - sin .times. .times. .theta. 2 )
.times. ( cos .times. .times. .theta. 2 + sin .times. .times.
.theta. 2 ) .times. r n 2 .times. cos .times. .times. .PHI.sin.PHI.
+ ( - 1 2 + cos .times. .times. .theta. 2 2 - sin .times. .times.
.theta. 2 ) 2 .times. r n 2 .times. cos 2 .times. .PHI. + ( cos
.times. .times. .theta. 2 - sin .times. .times. .theta. 2 ) 2
.times. r n 2 .times. sin 2 .times. .PHI. + 2 .times. ( - 1 2 + cos
.times. .times. .theta. 2 2 - sin .times. .times. .theta. 2 )
.times. ( cos .times. .times. .theta. 2 - sin .times. .times.
.theta. 2 ) .times. r n 2 .times. cos .times. .times. .PHI.sin.PHI.
+ ( - 1 2 + cos .times. .times. .theta. 2 ) 2 .times. r n 2 .times.
cos .times. 2 .times. .PHI. + sin 2 .times. .times. .theta. 2
.times. r n 2 .times. sin 2 .times. .PHI. + 2 .times. ( - 1 2 + cos
.times. .times. .theta. 2 ) .times. sin .times. .times. .theta. 2
.times. r n 2 .times. cos .times. .times. .PHI.sin.PHI. } ( 50 )
##EQU76##
[0120] Further multiplying out the squared terms gives x 2 + y 2 +
z 2 = r n 2 .times. { ( ( 1 2 + cos .times. .times. .theta. 2 2 ) 2
- ( 1 2 + cos .times. .times. .theta. 2 2 ) .times. .times. sin
.times. .times. .theta. + sin 2 .times. .times. .theta. 4 ) .times.
cos 2 .times. .PHI. + ( cos 2 .times. .theta. 2 + cos .times.
.times. .theta.sin .times. .times. .theta. 2 + sin 2 .times.
.theta. 4 ) .times. .times. sin 2 .times. .PHI. + 2 .times. ( 1 2 +
cos .times. .times. .theta. 2 2 - sin .times. .times. .theta. 2 )
.times. ( cos .times. .times. .theta. 2 - sin .times. .times.
.theta. 2 ) .times. cos .times. .times. .PHI.sin.PHI. + ( ( 1 2 +
cos .times. .times. .theta. 2 2 ) 2 + 1 2 + cos .times. .times.
.theta. 2 2 .times. sin .times. .times. .theta. + sin 2 .times.
.times. .theta. 4 ) .times. cos 2 .times. .PHI. + ( cos 2 .times.
.theta. 2 - cos .times. .times. .theta.sin.theta. 2 + sin 2 .times.
.theta. 4 ) 2 .times. sin 2 .times. .PHI. + 2 .times. ( - 1 2 + cos
.times. .times. .theta. 2 2 - sin .times. .times. .theta. 2 )
.times. ( cos .times. .times. .theta. 2 - sin .times. .times.
.theta. 2 ) .times. cos .times. .times. .PHI.sin.PHI. + ( 1 4 - cos
.times. .times. .theta. 2 + cos 2 .times. .theta. 4 ) .times.
.times. cos 2 .times. .PHI. + sin 2 .times. .theta. 2 .times. sin 2
.times. .PHI. + 2 .times. ( - 1 2 + cos .times. .times. .theta. 2 )
.times. sin .times. .times. .theta. 2 .times. cos .times. .times.
.PHI.sin.PHI. } ( 51 ) ##EQU77##
[0121] Combining terms gives x 2 + y 2 + z 2 = r n 2 .times. { ( 2
.times. ( 1 2 + cos .times. .times. .theta. 2 2 ) 2 + sin 2 .times.
.times. .theta. 2 + ( 1 4 - cos .times. .times. .theta. 2 + cos 2
.times. .theta. 4 ) ) .times. cos 2 .times. .PHI. + ( cos 2 .times.
.theta. + sin 2 .times. .times. .theta. 2 + sin 2 .times. .theta. 2
) .times. .times. sin 2 .times. .PHI. + 2 .times. ( 1 2 + cos
.times. .times. .theta. 2 2 - sin .times. .times. .theta. 2 )
.times. ( cos .times. .times. .theta. 2 + sin .times. .times.
.theta. 2 ) .times. cos .times. .times. .PHI.sin.PHI. + 2 .times. (
- 1 2 + cos .times. .times. .theta. 2 2 - sin .times. .times.
.theta. 2 ) .times. ( cos .times. .times. .theta. 2 - sin .times.
.times. .theta. 2 ) .times. cos .times. .times. .PHI.sin.PHI. + 2
.times. ( - 1 2 + cos .times. .times. .theta. 2 ) .times. sin
.times. .times. .theta. 2 .times. cos .times. .times. .PHI.sin.PHI.
} ( 52 ) ##EQU78##
[0122] Using the trigonometric identity from Eq. (46) and
multiplying out the trigonometric cross terms gives x 2 + y 2 + z 2
= r n 2 .times. { ( 1 4 + cos .times. .times. .theta. 2 + cos 2
.times. .theta. 4 + sin 2 .times. .theta. 2 + ( 1 4 - cos .times.
.times. .theta. 2 + cos 2 .times. .theta. 4 ) ) .times. .times. cos
2 .times. .PHI. + sin 2 .times. .PHI. + 2 .times. ( cos .times.
.times. .theta. 4 + cos 2 .times. .theta. 4 - sin .times. .times.
.theta.cos.theta. 2 .times. 2 + sin .times. .times. .theta. 4
.times. 2 + sin .times. .times. .theta.cos.theta. 4 .times. 2 - sin
2 .times. .times. .theta. 4 - cos .times. .times. .theta. 4 - cos 2
.times. .theta. 4 - sin .times. .times. .theta.cos.theta. 2 .times.
2 + sin .times. .times. .theta. 4 .times. 2 + sin .times. .times.
.theta.cos.theta. 4 .times. 2 + sin 2 .times. .times. .theta. 4 -
sin .times. .times. .theta. 2 .times. 2 + sin .times. .times.
.theta.cos.theta. 2 .times. 2 ) .times. cos .times. .times.
.PHI.sin.PHI. .times. } i . .times. ( 53 ) ##EQU79##
[0123] Combining terms gives x 2 + y 2 + z 2 = r n 2 .times. { ( 1
2 + cos 2 .times. .theta. 2 + sin 2 .times. .theta. 2 ) .times. cos
2 .times. .PHI. + sin 2 .times. .PHI. + 2 .times. ( 0 ) .times. cos
.times. .times. .PHI. .times. .times. sin .times. .times. .PHI. } (
54 ) ##EQU80##
[0124] By using the trigonometric identity of Eq. (46) for .theta.
and .phi., Eq. (54) becomes x.sup.2+y.sup.2+z.sup.2=r.sub.n.sup.2
(55)
[0125] which is the equation of a uniform sphere.
[0126] Component Orbitsphere-cvf Squared for STEP TWO by the
Rotation of a Great Circle About the ( - 1 2 .times. i x , 1 2
.times. i y , i z ) ##EQU81## -Axis by 2.pi.
[0127] From Eq. (22), the equation for the component
orbitsphere-cvf squared for STEP TWO is given by x 2 + y 2 + z 2 =
1 16 .times. r n 2 .times. { ( 2 .times. sin .times. .times.
.theta. .function. ( cos .times. .times. .PHI. - sin .times.
.times. .PHI. ) + 2 .times. ( 2 .times. cos .times. .times. .theta.
.times. .times. cos .times. .times. .PHI. + ( - 1 + cos .times.
.times. .theta. ) .times. sin .times. .times. .PHI. ) ) 2 + ( - 2
.times. sin .times. .times. .theta. .function. ( cos .times.
.times. .PHI. + sin .times. .times. .PHI. ) + 2 .times. ( cos
.times. .times. .theta. .function. ( 2 .times. cos .times. .times.
.PHI. - sin .times. .times. .PHI. ) + sin .times. .times. .PHI. ) )
2 + 4 .times. ( 2 .times. cos .times. .times. .PHI.sin .times.
.times. .theta. + ( 1 + cos .times. .times. .theta. ) .times. sin
.times. .times. .PHI. ) 2 } ( 56 ) ##EQU82##
[0128] Multiplying out the terms gives x 2 + y 2 + z 2 = 1 16
.times. r n 2 .times. { 4 .times. sin 2 .times. .theta. .times.
.times. ( cos .times. .times. .PHI. - sin .times. .times. .PHI. ) 2
+ 2 .times. ( 2 .times. cos .times. .times. .theta.cos .times.
.times. .PHI. + ( - 1 + cos .times. .times. .theta. ) .times.
.times. sin .times. .times. .PHI. ) 2 + .times. 4 .times. 2 .times.
sin .times. .times. .theta. .times. .times. ( cos .times. .times.
.PHI. - sin .times. .times. .PHI. ) .times. ( 2 .times. cos .times.
.times. .theta.cos.PHI. + ( - 1 + cos .times. .times. .theta. )
.times. .times. sin .times. .times. .PHI. ) + 4 .times. sin 2
.times. .theta. .times. .times. ( cos .times. .times. .PHI. + sin
.times. .times. .PHI. ) 2 + 2 .times. .times. ( cos .times. .times.
.theta. .times. .times. ( 2 .times. cos .times. .times. .PHI. - sin
.times. .times. .PHI. ) + sin .times. .times. .PHI. ) 2 - - 4
.times. 2 .times. sin .times. .times. .theta. .times. .times. ( cos
.times. .times. .PHI. + sin .times. .times. .PHI. ) .times. ( cos
.times. .times. .theta. .times. .times. ( 2 .times. cos .times.
.times. .PHI. - sin .times. .times. .PHI. ) + sin .times. .times.
.PHI. ) + .times. 8 .times. cos 2 .times. .PHI.sin 2 .times.
.theta. + 4 .times. .times. ( 1 + cos .times. .times. .theta. ) 2
.times. sin 2 .times. .PHI. + 8 .times. 2 .times. cos .times.
.times. .PHI.sin .times. .times. .theta. .times. .times. ( 1 + cos
.times. .times. .theta. ) .times. .times. sin .times. .times. .PHI.
.times. } ( 57 ) x 2 + y 2 + z 2 = 1 16 .times. r n 2 .times. { 4
.times. sin 2 .times. .theta. .times. .times. ( cos 2 .times.
.times. .PHI. - 2 .times. cos .times. .times. .PHI.sin .times.
.times. .PHI. + sin 2 .times. .PHI. ) + 2 .times. .times. ( 4
.times. cos 2 .times. .theta.cos 2 .times. .times. .PHI. + 4
.times. cos .times. .times. .theta.cos.PHI. .times. .times. ( - 1 +
cos .times. .times. .theta. ) .times. .times. sin .times. .times.
.PHI. + ( - 1 + cos .times. .times. .theta. ) 2 .times. .times. sin
2 .times. .times. .PHI. ) + 4 .times. 2 .times. sin .times. .times.
.theta. .times. .times. ( cos .times. .times. .PHI. - sin .times.
.times. .PHI. ) .times. ( 2 .times. .times. cos .times. .times.
.theta. .times. .times. cos .times. .times. .PHI. + ( - 1 + cos
.times. .times. .theta. ) .times. .times. sin .times. .times. .PHI.
) + 4 .times. sin 2 .times. .times. .theta. .times. .times. ( cos 2
.times. .times. .PHI. + 2 .times. cos .times. .times. .PHI.sin.PHI.
+ sin 2 .times. .PHI. ) + 2 .times. ( cos 2 .times. .theta.
.function. ( 2 .times. cos .times. .times. .PHI. - sin .times.
.times. .PHI. ) 2 + 2 .times. cos .times. .times. .theta. .times.
.times. ( 2 .times. cos .times. .times. .PHI. - sin .times. .times.
.PHI. ) .times. .times. sin .times. .times. .PHI. + sin 2 .times.
.PHI. ) - 4 .times. 2 .times. sin .times. .times. .theta. .times.
.times. ( cos .times. .times. .PHI. + sin .times. .times. .PHI. )
.times. ( cos .times. .times. .theta. .times. .times. ( 2 .times.
cos .times. .times. .PHI. - sin .times. .times. .PHI. ) + sin
.times. .times. .PHI. ) + .times. 8 .times. cos 2 .times. .PHI.sin
2 .times. .theta. + 4 .times. .times. ( 1 + 2 .times. cos .times.
.times. .theta. + cos 2 .times. .theta. ) .times. .times. sin 2
.times. .PHI. 8 .times. 2 .times. cos .times. .times. .PHI.sin
.times. .times. .theta. .times. .times. ( 1 + cos .times. .times.
.theta. ) .times. .times. sin .times. .times. .PHI. .times. } ii .
.times. ( 58 ) x 2 + y 2 + z 2 = 1 16 .times. r n 2 .times. { 4
.times. sin 2 .times. .theta.cos 2 .times. .PHI. - 8 .times. sin 2
.times. .theta.cos.PHI.sin.PHI. + 4 .times. sin 2 .times.
.theta.sin 2 .times. .PHI. + 8 .times. cos 2 .times. .theta.cos 2
.times. .PHI. - 8 .times. cos .times. .times. .theta.cos .times.
.times. .PHI.sin.PHI. + 8 .times. cos 2 .times.
.theta.cos.PHI.sin.PHI. + 2 .times. sin 2 .times. .PHI. - 4 .times.
cos .times. .times. .theta.sin 2 .times. .PHI. + 2 .times. cos 2
.times. .theta.sin 2 .times. .PHI. + 4 .times. 2 .times. sin
.times. .times. .theta. .times. .times. ( cos .times. .times. .PHI.
- sin .times. .times. .PHI. ) .times. ( 2 .times. cos .times.
.times. .theta.cos.PHI. - sin .times. .times. .PHI. + cos .times.
.times. .theta.sin .times. .times. .PHI. ) + 4 .times. sin 2
.times. .theta.cos 2 .times. .PHI. + 8 .times. sin 2 .times.
.theta.cos.PHI.sin.PHI. + 4 .times. sin 2 .times. .theta.sin 2
.times. .PHI. + 2 .times. cos 2 .times. .times. .theta. .times.
.times. ( 4 .times. cos 2 .times. .times. .PHI. - 4 .times. cos
.times. .times. .PHI.sin.PHI. + sin 2 .times. .PHI. ) + 8 .times.
cos .times. .times. .theta.cos.PHI.sin.PHI. - 4 .times. cos .times.
.times. .theta.sin 2 .times. .times. .PHI. + 2 .times. sin 2
.times. .PHI. - ( cos .times. .times. .PHI. + sin .times. .times.
.PHI. ) ( 8 .times. 2 .times. sin .times. .times.
.theta.cos.theta.cos.PHI. - 4 .times. 2 .times. sin .times. .times.
.theta.cos.theta.sin.PHI. + 4 .times. 2 .times. sin .times. .times.
.theta.sin.PHI. ) + 8 .times. cos 2 .times. .PHI.sin 2 .times.
.times. .theta. + 4 .times. sin 2 .times. .times. .PHI. + 8 .times.
sin 2 .times. .PHI.cos.theta. + 4 .times. sin 2 .times. .PHI.cos 2
.times. .PHI. + 8 .times. 2 .times. sin .times. .times.
.theta.cos.PHI.sin.PHI. + 8 .times. 2 .times. sin .times. .times.
.theta.cos.theta.cos.PHI.sin.PHI. } ii . .times. ( 59 ) x 2 + y 2 +
z 2 = 1 16 .times. r n 2 .times. { 4 .times. sin 2 .times.
.theta.cos 2 .times. .PHI. - 8 .times. sin 2 .times.
.theta.cos.PHI.sin.PHI. + 4 .times. sin 2 .times. .theta.sin 2
.times. .PHI. + 8 .times. cos 2 .times. .theta.cos 2 .times. .PHI.
- 8 .times. cos .times. .times. .theta.cos .times. .times.
.PHI.sin.PHI. + 8 .times. cos 2 .times. .theta.cos.PHI.sin.PHI. + 2
.times. sin 2 .times. .PHI. - 4 .times. cos .times. .times.
.theta.sin 2 .times. .PHI. + 2 .times. cos 2 .times. .theta.sin 2
.times. .PHI. + 8 .times. 2 .times. cos .times. .times. .theta.
.times. .times. sin .times. .times. .theta.cos 2 .times. .times.
.PHI. - 4 .times. 2 .times. sin .times. .times. .theta.cos.PHI.sin
.times. .times. .PHI. + 4 .times. 2 .times. cos .times. .times.
.theta.sin .times. .times. .theta.cos.PHI.sin.PHI. - 8 .times. 2
.times. cos .times. .times. .theta. .times. .times. sin .times.
.times. .theta.cos .times. .times. .PHI.sin.PHI. + 4 .times. 2
.times. sin .times. .times. .theta.sin 2 .times. .times. .PHI. - 4
.times. 2 .times. cos .times. .times. .theta.sin .times. .times.
.theta.sin 2 .times. .PHI. + 4 .times. sin 2 .times. .times.
.theta. .times. .times. cos 2 .times. .times. .PHI.8sin 2 .times.
.theta.cos .times. .times. .PHI.sin.PHI. + 4 .times. sin 2 .times.
.theta.sin 2 .times. .PHI. + 8 .times. cos .times. 2 .times.
.theta.cos 2 .times. .PHI. - 8 .times. cos .times. 2 .times.
.theta.cos .times. .times. .PHI.sin.PHI. + 2 .times. cos 2 .times.
.theta.sin 2 .times. .PHI. + 8 .times. cos .times. .times.
.theta.cos.PHI.sin.PHI. - 4 .times. cos .times. .times. .theta.sin
2 .times. .PHI. + 2 .times. sin 2 .times. .PHI. - 8 .times. 2
.times. cos .times. .times. .theta. .times. .times. sin .times.
.times. .theta.cos 2 .times. .times. .PHI. + 4 .times. 2 .times.
cos .times. .times. .theta.sin.theta.cos .times. .times.
.PHI.sin.PHI. - 4 .times. 2 .times. sin .times. .times.
.theta.cos.PHI.sin.PHI. - 8 .times. 2 .times. cos .times. .times.
.theta. .times. .times. sin .times. .times. .theta.cos .times.
.times. .PHI.sin .times. .times. .PHI. + 4 .times. 2 .times. cos
.times. .times. .theta.sin.theta.sin 2 .times. .times. .PHI. - 4
.times. 2 .times. sin .times. .times. .theta.sin 2 .times. .PHI. +
8 .times. sin 2 .times. .theta.cos 2 .times. .PHI. + 4 .times. sin
2 .times. .PHI. + 8 .times. cos .times. .times. .theta.sin 2
.times. .PHI. + 4 .times. cos 2 .times. .theta.sin 2 .times. .PHI.
+ 8 .times. 2 .times. sin .times. .times. .theta.cos.PHI.sin.PHI. +
8 .times. 2 .times. cos .times. .times.
.theta.sin.theta.cos.PHI.sin.PHI. } iv . .times. ( 60 ) x 2 + y 2 +
z 2 = 1 16 .times. r n 2 .times. { 4 .times. sin 2 .times.
.theta.cos 2 .times. .PHI. - 8 .times. sin 2 .times.
.theta.cos.PHI.sin.PHI. + 4 .times. sin 2 .times. .theta.sin 2
.times. .PHI. + 8 .times. cos 2 .times. .theta.cos 2 .times. .PHI.
- 8 .times. cos .times. .times. .theta.cos .times. .times.
.PHI.sin.PHI. + 8 .times. cos 2 .times. .theta.cos.PHI.sin.PHI. + 2
.times. sin 2 .times. .PHI. - 4 .times. cos .times. .times.
.theta.sin 2 .times. .PHI. + 2 .times. cos 2 .times. .theta.sin 2
.times. .PHI. + 8 .times. 2 .times. cos .times. .times. .theta.
.times. .times. sin .times. .times. .theta.cos 2 .times. .times.
.PHI. - 4 .times. 2 .times. sin .times. .times. .theta.cos.PHI.sin
.times. .times. .PHI. + 4 .times. 2 .times. cos .times. .times.
.theta.sin .times. .times. .theta.cos.PHI.sin.PHI. - 8 .times. 2
.times. cos .times. .times. .theta. .times. .times. sin .times.
.times. .theta.cos .times. .times. .PHI.sin.PHI. + 4 .times. 2
.times. sin .times. .times. .theta.sin 2 .times. .times. .PHI. - 4
.times. 2 .times. cos .times. .times. .theta.sin .times. .times.
.theta.sin 2 .times. .PHI. + 4 .times. sin 2 .times. .times.
.theta. .times. .times. cos 2 .times. .times. .PHI. + 8 .times. sin
2 .times. .theta.cos .times. .times. .PHI.sin.PHI. + 4 .times. sin
2 .times. .theta.sin 2 .times. .PHI. + 8 .times. cos .times. 2
.times. .theta.cos 2 .times. .PHI. - 8 .times. cos .times. 2
.times. .theta.cos .times. .times. .PHI.sin.PHI. + 2 .times. cos 2
.times. .theta.sin 2 .times. .PHI. + 8 .times. cos .times. .times.
.theta.cos.PHI.sin.PHI. - 4 .times. cos .times. .times. .theta.sin
2 .times. .PHI. + 2 .times. sin 2 .times. .PHI. - 8 .times. 2
.times. cos .times. .times. .theta. .times. .times. sin .times.
.times. .theta.cos 2 .times. .times. .PHI. + 4 .times. 2 .times.
cos .times. .times. .theta.sin.theta.cos .times. .times.
.PHI.sin.PHI. - 4 .times. 2 .times. sin .times. .times.
.theta.cos.PHI.sin.PHI. - 8 .times. 2 .times. cos .times. .times.
.theta. .times. .times. sin .times. .times. .theta.cos .times.
.times. .PHI.sin .times. .times. .PHI. + 4 .times. 2 .times. cos
.times. .times. .theta.sin.theta.sin 2 .times. .times. .PHI. - 4
.times. 2 .times. sin .times. .times. .theta.sin 2 .times. .PHI. +
8 .times. sin 2 .times. .theta.cos 2 .times. .PHI. + 4 .times. sin
2 .times. .PHI. + 8 .times. cos .times. .times. .theta.sin 2
.times. .PHI. + 4 .times. cos 2 .times. .theta.sin 2 .times. .PHI.
+ 8 .times. 2 .times. sin .times. .times. .theta.cos.PHI.sin.PHI. +
8 .times. 2 .times. cos .times. .times.
.theta.sin.theta.cos.PHI.sin.PHI. } v . .times. ( 61 )
##EQU83##
[0129] Combining terms gives x 2 + y 2 + z 2 = 1 16 .times. r n 2
.times. { 4 .times. sin 2 .times. .theta. .times. .times. sin 2
.times. .PHI. + 4 .times. sin 2 .times. .theta. .times. .times. sin
2 .times. .PHI. + 4 .times. sin 2 .times. .theta. .times. .times.
cos 2 .times. .PHI. + 8 .times. .times. sin 2 .times. .theta.
.times. .times. cos 2 .times. .PHI. + 4 .times. sin 2 .times.
.theta. .times. .times. cos 2 .times. .PHI. + 4 .times. cos 2
.times. .theta. .times. .times. sin 2 .times. .PHI. + 2 .times. cos
2 .times. .theta. .times. .times. sin 2 .times. .PHI. + 2 .times.
cos 2 .times. .theta. .times. .times. sin 2 .times. .PHI. + 8
.times. cos 2 .times. .theta. .times. .times. cos 2 .times. .PHI. +
8 .times. cos 2 .times. .theta. .times. .times. cos 2 .times. .PHI.
+ 2 .times. sin 2 .times. .PHI. + 2 .times. sin 2 .times. .PHI. + 4
.times. sin 2 .times. .PHI. - 8 .times. sin 2 .times. .theta.
.times. .times. cos .times. .times. .PHI. .times. .times. sin
.times. .times. .PHI. + 8 .times. sin 2 .times. .theta. .times.
.times. cos .times. .times. .PHI. .times. .times. sin .times.
.times. .PHI. + 8 .times. cos 2 .times. .theta. .times. .times. cos
.times. .times. .PHI. .times. .times. sin .times. .times. .PHI. - 8
.times. cos 2 .times. .times. .theta. .times. .times. cos .times.
.times. .PHI. .times. .times. sin .times. .times. .PHI. - 4 .times.
cos .times. .times. .theta. .times. .times. sin 2 .times. .PHI. - 4
.times. cos .times. .times. .theta. .times. .times. sin 2 .times.
.PHI. + 8 .times. cos .times. .times. .theta. .times. .times. sin 2
.times. .PHI. - 8 .times. cos .times. .times. .theta. .times.
.times. cos .times. .times. .PHI. .times. .times. sin .times.
.times. .PHI. + 8 .times. cos .times. .times. .theta. .times.
.times. cos .times. .times. .PHI. .times. .times. sin .times.
.times. .PHI. - 8 .times. 2 .times. cos .times. .times. .theta.
.times. .times. sin .times. .times. .theta. .times. .times. cos 2
.times. .PHI. + 8 .times. 2 .times. cos .times. .times. .theta.
.times. .times. sin .times. .times. .theta. .times. .times. cos 2
.times. .PHI. - 4 .times. 2 .times. cos .times. .times. .theta.
.times. .times. sin .times. .times. .theta. .times. .times. sin 2
.times. .PHI. + 4 .times. 2 .times. cos .times. .times. .theta.
.times. .times. sin .times. .times. .theta. .times. .times. sin 2
.times. .PHI. - 4 .times. 2 .times. sin .times. .times. .theta.
.times. .times. cos .times. .times. .PHI. .times. .times. sin
.times. .times. .PHI. - 4 .times. 2 .times. sin .times. .times.
.theta. .times. .times. cos .times. .times. .PHI.sin .times.
.times. .PHI. + 8 .times. 2 .times. sin .times. .times. .theta.
.times. .times. cos .times. .times. .PHI. .times. .times. sin
.times. .times. .PHI. - 8 .times. 2 .times. cos .times. .times.
.theta. .times. .times. sin .times. .times. .theta. .times. .times.
cos .times. .times. .PHI. .times. .times. sin .times. .times. .PHI.
+ 8 .times. 2 .times. cos .times. .times. .theta. .times. .times.
sin .times. .times. .theta. .times. .times. cos .times. .times.
.PHI. .times. .times. sin .times. .times. .PHI. + 4 .times. 2
.times. cos .times. .times. .theta. .times. .times. sin .times.
.times. .theta. .times. .times. cos .times. .times. .PHI.sin
.times. .times. .PHI. + 4 .times. 2 .times. cos .times. .times.
.theta. .times. .times. sin .times. .times. .theta. .times. .times.
cos .times. .times. .PHI. .times. .times. sin .times. .times. .PHI.
- 8 .times. 2 .times. cos .times. .times. .theta. .times. .times.
sin .times. .times. .theta. .times. .times. cos .times. .times.
.PHI. .times. .times. sin .times. .times. .PHI. + 4 .times. 2
.times. sin .times. .times. .theta. .times. .times. sin 2 .times.
.PHI. - 4 .times. 2 .times. sin .times. .times. .theta. .times.
.times. sin 2 .times. .PHI. } .times. vi . ( 62 ) x 2 + y 2 + z 2 =
1 16 .times. r n 2 .times. { 8 .times. sin 2 .times. .theta.
.times. .times. sin 2 .times. .PHI. + 16 .times. sin 2 .times.
.theta. .times. .times. cos 2 .times. .PHI. + 8 .times. cos 2
.times. .theta. .times. .times. sin 2 .times. .PHI. + 16 .times.
cos 2 .times. .theta. .times. .times. cos 2 .times. .PHI. + 8
.times. sin 2 .times. .PHI. } ( 63 ) ##EQU84##
[0130] By using the trigonometric identity of Eq. (46) for .theta.
and .phi., Eq. (63) becomes x 2 + y 2 + z 2 = 1 16 .times. r n 2
.times. { 8 .times. sin 2 .times. .PHI. .function. ( sin 2 .times.
.theta. + cos 2 .times. .theta. ) + 16 .times. cos 2 .times. .PHI.
.function. ( sin 2 .times. .theta. + cos 2 .times. .theta. ) + 8
.times. sin 2 .times. .PHI. } ( 64 ) x 2 + y 2 + z 2 = 1 16 .times.
r n 2 .times. { 16 .times. ( sin 2 .times. .PHI. + cos 2 .times.
.PHI. ) } ( 65 ) x 2 + y 2 + z 2 = r n 2 ( 66 ) ##EQU85##
[0131] which is the equation of a uniform sphere.
[0132] Orbitsphere-cvf Squared
[0133] Eqs. (48). (55), and (66) are each the equation of a uniform
sphere. The superposition of the uniform distributions from STEP
ONE and STEP TWO is the uniform current density function
Y.sub.0.sup.0(.theta.,.phi.) that is an equipotential, minimum
energy surface shown in FIG. 13. The angular momentum is
identically that from the superposition of the primary component
orbitsphere-cvfs of the orbitsphere-cvf, L xy = 4 .times. .times.
and .times. .times. L z = 2 . ##EQU86## The spatially uniform
electron current having the orthogonal angular momentum components
given by Eqs. (1.76-1.77) of Mills GUT can then be considered
conceptually from two viewpoints regarding the basis element of the
orbitsphere-cvf which is a two-dimensional vector field comprised
of an infinite number of one-dimensional great circles having
zero-dimensional crossings. The electron current,
Y.sub.0.sup.0(.theta.,.phi.), is a continuous uniform superposition
of secondary orbitsphere-cvfs onto and over the two-dimensional
surface wherein each secondary orbitsphere-cvf of equivalent
angular momentum, orientation, and phase replaces a corresponding
great-circle current loops of the primary orbitsphere-cvf. Or,
equivalently, the primary orbitsphere-cvf is the compression of a
secondary two-dimensional orbitsphere-cvf into each of the infinite
number of one-dimensional great circles such that L.sub.R, the
orientation, and the phase of the former element matches that of
the latter over a two-dimensional spherical shell to form a primary
two-dimensional vector field.
[0134] Matrices to Demonstrate the Convolution to Generate the
Uniform Current (Charge)-Density Function
Y.sub.0.sup.0(.theta.,.phi.)
[0135] Either one of the orthogonal basis-element great circles
generates the component orbitsphere-cvf for STEP ONE and STEP TWO
as given in the STEP ONE by the Rotation of a Great Circle about
the (i.sub.x,i.sub.y,0i.sub.z)-Axis by 2.pi. section and the STEP
TWO by Rotation of a Great Circle About ( - 1 2 .times. i x , 1 2
.times. i y , i z ) ##EQU87## -Axis by 2.pi. section, respectively.
Thus, either of the STEP ONE and the STEP TWO components can serve
as the primary component orbitsphere-cvf for the convolution
operation given by Eq. (38). Similarly, either one of the
orthogonal basis-element great circles generates the component
orbitsphere-cvf having a stationary angular momentum vector
associated with STEP ONE and STEP TWO as given in the
Orbitsphere-cvf Orthogonal to that of STEP ONE by the Rotation of a
Great Circle about the (i.sub.x,-i.sub.y,0i.sub.z)-Axis by 2.pi.
section and the STEP TWO by Rotation of a Great Circle About the (
- 1 2 .times. i x , 1 2 .times. i y , i z ) ##EQU88## -Axis by
2.pi. section, respectively. Thus, either of the STEP ONE and the
STEP TWO-associated components can serve as the secondary component
orbitsphere-cvf for the convolution operation given by Eq. (38).
Computer modeling of the analytical equations to generate the
orbitsphere current vector field and the uniform current (charge)
density function Y.sub.0.sup.0(.theta.,.phi.) is available on the
web [3].
[0136] STEP ONE Matrices to Visualize the Currents of
Y.sub.0.sup.0(.theta.,.phi.)
[0137] Consider the case that the STEP-ONE primary component
orbitsphere-cvf is given by Eqs. (4) and (5) and the
STEP-ONE-associated secondary component orbitsphere-cvf is given by
Eqs. (30-33). The basis-element great circle of the primary
component orbitsphere-cvf is in the yz-plane as shown in FIG. 2,
and the current is counterclockwise. Thus, the angular momentum is
along the x-axis. The angular-momentum-and-orientation-matched
secondary component orbitsphere-cvf is shown in FIG. 10 and is
generated by Eq. (30). In this case, the secondary component
orbitsphere-cvf is aligned on the yz-plane and the resultant
angular momentum vector, L.sub.R, of the secondary component
orbitsphere-cvf is also along the x-axis.
[0138] Then, the uniform current distribution is given from Eq.
(38) as a infinite sum of the convolved elements comprising the
secondary component orbitsphere-cvf given by Eq. (30) rotated
according to Eq. (4), the matrix which generated the primary
component orbitsphere-cvf. The resulting constant function is exact
as given by Eq. (48). A representation that shows the current
elements can be generated by showing the basis-element secondary
component orbitsphere-cvf as a sum of great circles using Eq. (30)
and by showing the continuous convolution as a sum of discrete
incremental rotations of the position of the secondary component
orbitsphere-cvf using Eq. (4). In the case that the discrete
representation of the secondary component orbitsphere-cvf comprises
N great circles and the number of convolved secondary component
orbitsphere-cvf elements is M, the representation of the uniform
current density function showing current loops shown in FIGS. 14
and 15 is given by [ x ' y ' z ' ] = m = 1 m = M .times. [ 1 2 +
cos .function. ( m .times. .times. 2 .times. .pi. M ) 2 1 2 - cos
.function. ( m .times. .times. 2 .times. .pi. M ) 2 - sin
.function. ( m .times. .times. 2 .times. .pi. M ) 2 1 2 - cos
.function. ( m .times. .times. 2 .times. .pi. M ) 2 1 2 + cos
.function. ( m .times. .times. 2 .times. .pi. 2 ) 2 sin .function.
( m .times. .times. 2 .times. .pi. M ) 2 sin .function. ( m .times.
.times. 2 .times. .pi. M ) 2 - sin .function. ( m .times. .times. 2
.times. .pi. M ) 2 cos .function. ( m .times. .times. 2 .times.
.pi. M ) ] n = 1 n = N .times. [ cos .function. ( .pi. 4 ) - sin
.function. ( .pi. 4 ) 0 sin .function. ( .pi. 4 ) .times. cos
.function. ( n .times. .times. 2 .times. .pi. N ) cos .function. (
.pi. 4 ) .times. cos .function. ( n .times. .times. 2 .times. .pi.
N ) sin .function. ( n .times. .times. 2 .times. .pi. N ) - sin
.function. ( .pi. 4 ) .times. sin .function. ( n .times. .times. 2
.times. .pi. N ) - cos .function. ( .pi. 4 ) .times. sin .function.
( n .times. .times. 2 .times. .pi. N ) cos .function. ( n .times.
.times. 2 .times. .pi. N ) ] .times. [ 0 r n .times. cos .times.
.times. .PHI. r n .times. sin .times. .times. .PHI. ] ( 67 )
##EQU89##
[0139] The solution for the discrete form of
Y.sub.0.sup.0(.theta.,.phi.), the uniform, minimum-energy,
equipotential-energy function of the electron, given by Eq. (67) is
the basis of the uniform current-density structure and
architectural structure of the present Invention comprised of
great-circle elements. The latter is called a great dome wherein
the stresses are preferably equally distributed over the surface
under pure compression.
[0140] STEP TWO Matrices to Visualize the Currents of
Y.sub.0.sup.0(.theta.,.phi.)
[0141] Discrete Convolution with a Secondary Component
Orbitsphere-cvf in a Plane Along the (i.sub.x,i.sub.y,0i.sub.z)-
and z-Axes (xyz-Plane)
[0142] The resultant angular momentum vector, L.sub.R, of the
secondary component orbitsphere-cvf given by Eq. (32) is along
(i.sub.x,-i.sub.y,0i.sub.z), corresponding to a basis-element great
circle in the yz-plane having a counterclockwise current. The
angular momentum direction is reversed by reversing the direction
of the current to clockwise. Consider the case that the STEP-TWO
primary component orbitsphere-cvf is given by Eqs. (20-23).
Further, consider the case that the STEP-TWO-associated secondary
component orbitsphere-cvf is given by Eq. (32). The basis-element
great circle of the primary component orbitsphere-cvf shown in FIG.
7 is in the xyz-plane, and the current is clockwise. Thus, the
angular momentum is along the (-i.sub.x,i.sub.y,0i.sub.z)-axis. The
angular-momentum-and-orientation-matched secondary component
orbitsphere-cvf is shown in FIG. 11 and is generated by Eq. (32).
In this case, the secondary component orbitsphere-cvf is aligned on
the xyz-plane and the resultant angular momentum vector, L.sub.R,
of the secondary component orbitsphere-cvf is also along the
(-i.sub.x,i.sub.y,0i.sub.z)-axis.
[0143] Then, the uniform current distribution is given from Eq.
(38) as a infinite sum of the convolved elements comprising the
secondary component orbitsphere-cvf given by Eq. (32) rotated
according to Eq. (20), the matrix which generated the primary
component orbitsphere-cvf. The resulting constant function is exact
as given by Eq. (66). A representation that shows the current
elements can be generated by showing the basis-element secondary
component orbitsphere-cvf as a sum of great circles using Eq. (32)
and by showing the continuous convolution as a sum of discrete
incremental rotations of the position of the secondary component
orbitsphere-cvf using Eq. (20). In the case that the discrete
representation of the secondary component orbitsphere-cvf comprises
N great circles and the number of convolved secondary component
orbitsphere-cvf elements is M, the representation of the uniform
current density function showing current loops shown in FIGS. 16-18
is given by [ x ' y ' z ' ] = .times. .times. m = 1 m = M .times.
.times. [ 1 4 .times. ( 1 + 3 .times. cos .function. ( m .times.
.times. 2 .times. .pi. M ) ) 1 4 .times. ( - 1 + cos .times. ( m
.times. .times. 2 .times. .pi. M ) + 2 .times. 2 .times. sin
.function. ( m .times. .times. 2 .times. .pi. M ) ) 1 4 .times. ( -
2 + 2 .times. cos .function. ( m .times. .times. 2 .times. .pi. M )
- 2 .times. sin .function. ( m .times. .times. 2 .times. .pi. M ) )
1 4 .times. ( - 1 + cos .function. ( m .times. .times. 2 .times.
.pi. M ) - 2 .times. 2 .times. sin .function. ( m .times. .times. 2
.times. .pi. M ) ) 1 4 .times. ( 1 + 3 .times. cos .function. ( m
.times. .times. 2 .times. .pi. M ) ) 1 4 .times. ( 2 - 2 .times.
cos .function. ( m .times. .times. 2 .times. .pi. M ) - 2 .times.
sin .function. ( m .times. .times. 2 .times. .pi. M ) ) 1 2 .times.
( - 1 + cos .function. ( m .times. .times. 2 .times. .pi. M ) 2 +
sin .function. ( m .times. .times. 2 .times. .pi. M ) ) 1 4 .times.
( 2 - 2 .times. cos .function. ( m .times. .times. 2 .times. .pi. M
) + 2 .times. sin .function. ( m .times. .times. 2 .times. .times.
.pi. M ) ) cos 2 .times. ( m .times. .times. 2 .times. .pi. M ) 2 ]
n = 1 n = N .times. [ 1 2 + cos .function. ( n .times. .times. 2
.times. .pi. N ) 2 - 1 2 + cos .function. ( n .times. .times. 2
.times. .pi. N ) 2 sin .function. ( n .times. .times. 2 .times.
.pi. N ) 2 - 1 2 + cos .function. ( n .times. .times. 2 .times.
.pi. N ) 2 1 2 + cos .function. ( n .times. .times. 2 .times. .pi.
N ) 2 sin .function. ( n .times. .times. 2 .times. .pi. N ) 2 - sin
.function. ( n .times. .times. 2 .times. .pi. N ) 2 - sin
.function. ( n .times. .times. 2 .times. .pi. N ) 2 cos .function.
( n .times. .times. 2 .times. .pi. N ) ] .times. [ 0 r n .times.
cos .times. .times. .PHI. r n .times. sin .times. .times. .PHI. ] (
68 ) ##EQU90##
[0144] Discrete Convolution with a Secondary Component
Orbitsphere-cvf in the xy-Plane
[0145] Consider the case that the STEP-TWO primary component
orbitsphere-cvf is given by Eqs. (20-23). Further, consider the
case that the STEP-TWO associated secondary component
orbitsphere-cvf is from Eq. (7). The basis-element great circle of
the primary component orbitsphere-cvf is in the xy-plane as shown
in FIG. 6, and the current is counterclockwise. Thus, the angular
momentum is along the z-axis. The
angular-momentum-and-orientation-matched secondary component
orbitsphere-cvf is shown in FIG. 4 and is generated by Eq. (7). In
order to match phase, Eq. (7) must be rotated about the z-axis by
.pi. 4 ##EQU91## using zrot .function. ( .pi. 4 ) ##EQU92## using
Eq. (11). In this case, the secondary component orbitsphere-cvf is
aligned on the xy-plane and the resultant angular momentum vector,
L.sub.R, of the secondary component orbitsphere-cvf is also along
the z-axis. It is given by [ x ' y ' z ' ] = zrot .function. ( .pi.
4 ) [ cos .function. ( .pi. 4 ) .times. cos .times. .times. .theta.
sin .times. .times. .theta. sin .function. ( .pi. 4 ) .times. cos
.times. .times. .theta. - cos .function. ( .pi. 4 ) .times. sin
.times. .times. .theta. cos .times. .times. .theta. - sin
.function. ( .pi. 4 ) .times. sin .times. .times. .theta. - sin
.function. ( .pi. 4 ) 0 cos .function. ( .pi. 4 ) ] .function. [ r
n .times. cos .times. .times. .PHI. r n .times. sin .times. .times.
.PHI. 0 ] ( 69 ) ##EQU93##
[0146] Substitution of Eq. (11) into Eq. (69) gives [ x ' y ' z ' ]
= [ cos .times. .times. .theta. 2 - sin .times. .times. .theta. 2
sin .times. .times. .theta. 2 + cos .times. .times. .theta. 2 cos
.times. .times. .theta. 2 - sin .times. .times. .theta. 2 - cos
.times. .times. .theta. 2 - sin .times. .times. .theta. 2 - sin
.times. .times. .theta. 2 + cos .times. .times. .theta. 2 - cos
.times. .times. .theta. 2 - sin .times. .times. .theta. 2 - 1 2 0 1
2 ] .function. [ r n .times. cos .times. .times. .PHI. r n .times.
sin .times. .times. .PHI. 0 ] ( 70 ) ##EQU94##
[0147] Then, the uniform current distribution is given from Eq.
(38) as a infinite sum of the convolved elements comprising the
secondary component orbitsphere-cvf given by Eq. (70) rotated
according to Eq. (20), the matrix which generated the primary
component orbitsphere-cvf using Eq. (23) with Eq. (20). The
resulting constant function is exact as given by Eq. (66). A
representation that shows the current elements can be generated by
showing the basis-element secondary component orbitsphere-cvf as a
sum of great circles using Eq. (70) by showing the continuous
convolution as a sum of discrete incremental rotations of the
position of the secondary component orbitsphere-cvf using Eq. (20).
In the case that the discrete representation of the secondary
component orbitsphere-cvf comprises N great circles and the number
of convolved secondary component orbitsphere-cvf elements is M, the
representation of the uniform current density function showing
current loops shown in FIGS. 19-21 is given by [ x ' y ' z ' ] =
.times. .times. m = 1 m = M .times. [ 1 4 .times. ( 1 + 3 .times.
cos .function. ( m .times. .times. 2 .times. .pi. M ) ) 1 4 .times.
( - 1 + cos .times. ( m .times. .times. 2 .times. .pi. M ) + 2
.times. 2 .times. sin .function. ( m .times. .times. 2 .times. .pi.
M ) ) 1 4 .times. ( - 2 + 2 .times. cos .function. ( m .times.
.times. 2 .times. .pi. M ) - 2 .times. sin .function. ( m .times.
.times. 2 .times. .pi. M ) ) 1 4 .times. ( - 1 + cos .function. ( m
.times. .times. 2 .times. .pi. M ) - 2 .times. 2 .times. sin
.function. ( m .times. .times. 2 .times. .pi. M ) ) 1 4 .times. ( 1
+ 3 .times. cos .function. ( m .times. .times. 2 .times. .pi. M ) )
1 4 .times. ( 2 - 2 .times. cos .function. ( m .times. .times. 2
.times. .pi. M ) - 2 .times. sin .function. ( m .times. .times. 2
.times. .pi. M ) ) 1 2 .times. ( - 1 + cos .function. ( m .times.
.times. 2 .times. .pi. M ) 2 + sin .function. ( m .times. .times. 2
.times. .pi. M ) ) 1 4 .times. ( 2 - 2 .times. cos .function. ( m
.times. .times. 2 .times. .pi. M ) + 2 .times. sin .function. ( m
.times. .times. 2 .times. .times. .pi. M ) ) cos 2 .times. ( m
.times. .times. 2 .times. .pi. M ) 2 ] n = 1 n = N .times. [ cos
.function. ( n .times. .times. 2 .times. .pi. N ) 2 - sin
.function. ( n .times. .times. 2 .times. .pi. N ) 2 sin .function.
( n .times. .times. 2 .times. .pi. N ) 2 + cos .function. ( n
.times. .times. 2 .times. .pi. N ) 2 cos .function. ( n .times.
.times. 2 .times. .pi. N ) 2 - sin .function. ( n .times. .times. 2
.times. .pi. N ) 2 - cos .function. ( n .times. .times. 2 .times.
.pi. N ) 2 - sin .function. ( n .times. .times. 2 .times. .pi. N )
2 - sin .function. ( n .times. .times. 2 .times. .pi. N ) 2 + cos
.function. ( n .times. .times. 2 .times. .pi. N ) 2 - cos
.function. ( n .times. .times. 2 .times. .pi. N ) 2 - sin
.function. ( n .times. .times. 2 .times. .pi. N ) 2 - 1 2 0 1 2 ]
.function. [ r n .times. cos .times. .times. .PHI. r n .times. sin
.times. .times. .PHI. 0 ] ( 71 ) ##EQU95##
* * * * *
References