U.S. patent application number 12/454868 was filed with the patent office on 2010-12-02 for method and apparatus for indicating rotational speed of footballs.
Invention is credited to Raffie Ashot Karabed, Razmik Karabed.
Application Number | 20100304905 12/454868 |
Document ID | / |
Family ID | 43220899 |
Filed Date | 2010-12-02 |
United States Patent
Application |
20100304905 |
Kind Code |
A1 |
Karabed; Raffie Ashot ; et
al. |
December 2, 2010 |
Method and apparatus for indicating rotational speed of
footballs
Abstract
Indicia on the surface of a football (for the American variety
of the sport) are provided to provide visual means for assessing
the spin speed of the football when thrown in a desired manner. In
one embodiment, the indicia comprise a circumferential indicator
ring consisting of a plurality of alternatingly-colored segments,
the indicator ring being aligned with the desired direction of spin
of the football. The segments are preferably selected to be of
contrasting colors, and the width of the segments is selected such
that the indicator ring appears to an observer to be a single,
mixed color provided the football is spinning about its axis at at
least a predetermined frequency. In another embodiment, the indicia
comprise two or more circumferential indicator rings at least one
of which comprising a plurality of alternatingly-colored segments.
The respective segment widths on one or more rings may be selected
such that the mixed color of one ring, and hence the spin speed of
the football, can be assessed relative to the color or mixed color
of another ring.
Inventors: |
Karabed; Raffie Ashot; (San
Jose, CA) ; Karabed; Razmik; (San Jose, CA) |
Correspondence
Address: |
Hugh R. Kress;Adair & Myers PLLC
Suite 320, 3120 Southwest Freeway
Houston
TX
77098
US
|
Family ID: |
43220899 |
Appl. No.: |
12/454868 |
Filed: |
May 26, 2009 |
Current U.S.
Class: |
473/603 |
Current CPC
Class: |
A63B 43/002 20130101;
A63B 2243/007 20130101; A63B 43/008 20130101 |
Class at
Publication: |
473/603 |
International
Class: |
A63B 41/08 20060101
A63B041/08 |
Claims
1. A football adapted to be thrown in a manner resulting in
spinning of said football substantially around a long axis thereof
as said football travels along a trajectory; said football having
spin speed indicia applied on an outer surface, said indicia being
aligned with the direction of spin of said football about said long
axis; wherein said indicia provide an indication of whether said
football is spinning at at least a first predetermined minimum spin
speed.
2. The football of claim 1, wherein said indicia comprise a first
circumferential indicator ring on said outer surface.
3. The football of claim 2, wherein said first circumferential
indicator ring comprises a plurality of alternatingly-colored
segments of a first color and a second color, said segments forming
a mixed color when said football spins at at least said first
predetermined minimum speed.
4. The football of claim 3, wherein the width of said segments is
selected to establish a said first predetermined minimum spin
speed.
5. The football of claim 31, wherein said indicia further comprise
a second circumferential indicator ring on said outer surface.
6. The football of claim 5, wherein said second indicator ring
provides a single reference color relative to said mixed color of
said first indicator ring.
7. The football of claim 6, wherein said second indicator ring
comprises a plurality of alternatingly-colored segments.
8. The football of claim 6, wherein said second indicator ring
comprises a plurality of alternatingly-colored segments of said
first color and said second color.
9. The football of claim 7, wherein said alternatingly-colored
segments of said second indicator ring have a different width then
said alternatingly-colored segments of said first indicator
ring.
10. The football of claim 7, wherein said second indicator ring
appears to be a single, mixed color when said football spins at at
least a second predetermined minimum spin speed, said second
predetermined minimum spin speed being less than said first
predetermined minimum spin speed.
11. A method of providing a visual indication of the spin speed of
a football adapted to be thrown in a manner resulting in spinning
of said football substantially around a long axis thereof as said
football travels along a trajectory, the method comprising:
applying spin speed indicia on an outer surface of said football,
said indicia being aligned with the direction of spin of said
football about said long axis. wherein said indicia provide an
indication of whether said football is spinning at at least a first
predetermined minimum spin speed.
12. The method of claim 11, wherein said indicia comprise a first
circumferential indicator ring on said outer surface.
13. The method of claim 12, wherein said first circumferential
indicator ring comprises a plurality of alternatingly-colored
segments of a first color and a second color, said segments forming
a mixed color when said football spins at at least said first
predetermined minimum speed.
14. The method of claim 13, wherein the width of said segments is
selected to establish a said first predetermined minimum spin
speed.
15. The method of claim 13, further comprising applying a second
circumferential indicator ring on said outer surface.
16. The method of claim 15, wherein said second indicator ring
provides a single reference color relative to said mixed color of
said first indicator ring.
17. The method of claim 16, wherein said second indicator ring
comprises a plurality of alternatingly-colored segments.
18. The method of claim 16, wherein said second indicator ring
comprises a plurality of alternatingly-colored segments of said
first color and said second color.
19. The method of claim 17, wherein said alternatingly-colored
segments of said second indicator ring have a different width then
said alternatingly-colored segments of said first indicator
ring.
20. The method of claim 7, wherein said second indicator ring
appears to be a single, mixed color when said football spins at at
least a second predetermined minimum spin speed, said second
predetermined minimum spin speed being less than said first
predetermined minimum spin speed.
Description
FIELD OF THE INVENTION
[0001] The present invention relates generally to footballs having
a generally prolate spheroid shape, and more particularly relates
to indicating the rotational speed of a thrown football by means of
a visual indicator.
BACKGROUND OF THE INVENTION
[0002] In the American, Canadian, and Australian varieties of the
game of football, as well as in the game of rugby, a ball having a
generally prolate spheroid shape is used. (Australian footballs and
rugby footballs tend to have more rounded ends than American and
Canadian footballs. As used herein, the term "football" shall be
understood to refer to any ball having a generally prolate spheroid
shape, regardless of the particular sport for which it is
intended). To produce a smooth and efficient trajectory, a football
is preferably thrown such that it rotates or spins about its long
axis (i.e., the axis through the ball's pointed ends). This spin is
understood to generate forces that minimize wobbling of the ball,
leading to a smooth, accurate, and energy-efficient trajectory. A
professional football player can give a football a spin speed of
about 10 revolutions per second. See, e.g., Wattsa et al., "The
Drag Force on an American Football," Am. J. Phys., Vol. 71, No. 8;
August 2003.
[0003] In general, throwing a football accurately and efficiently
is not an easily learned skill. A portion of the necessary skill
involves achieving rapid spinning of the ball about its long axis.
Both a beginner quarterback and a well skilled one are concerned
with the spin speed of the ball. Consequently, for the purpose of
developing and refining such skill, it would be advantageous to
have a means of measuring the spin speed of a football during its
flight.
[0004] Heretofore, there have not been shown any practical means of
indicating or measuring the spin speed of a thrown football. More
specifically, it is believed by the inventors that the prior art
provides no practical methods for one to measure a football's spin
speed or to evaluate progress one has made in achieving desired
levels of spin on a football.
[0005] It is known in the prior art to provide multiple colors and
other indicia on the surface of a football. However, it is believed
that this is typically done primarily or exclusively for ornamental
or cosmetic reasons, and not to provide information about a desired
spin speed.
SUMMARY OF THE INVENTION
[0006] In view of the foregoing, the present invention is directed
to a method and apparatus for indicating the rate of spin of a
thrown football.
[0007] In accordance with one embodiment of the invention, indicia
in the form of one or more indicator rings, comprising segments of
alternating colors, are applied in some manner to the surface of a
football. The indicia are functionally aligned with the desired
spin direction of the ball. In one embodiment, the colors of an
indicator ring and the widths of an indicator ring's
alternatingly-colored segments are selected so that if the ball is
spinning faster than a nominal speed, then the two colors of the
ring appear to an observer to be one color, under nominal outdoor
light conditions. In this disclosure, the term "mixed color" is
used to refer to the observed single color created by the spinning
of an indicator ring applied to a spinning object such as a
football.
[0008] In accordance with one aspect of the invention, if an
indicator ring applied to a football is observed to be
substantially a single mixed color, this confirms to an observer
that the spin speed of the football must be faster than a nominal
speed.
[0009] In accordance with another aspect of the invention, a
practicing quarterback can assess his/her progress in perfecting
passing skills by assessing whether a football has been thrown with
sufficient spin as to cause an indicator ring to appear to be one
color instead of two colors.
[0010] In accordance with another aspect of the invention, a
football having a two-color indicator ring as disclosed herein
provides a means for both beginning and advanced quarterbacks to
consistently evaluate spin speed and thereby assess his/her
progress in developing passing skills.
[0011] In accordance with one aspect of the invention, the indicia
on the surface of a football are provided to provide visual means
for assessing the spin speed of the football when thrown in a
desired manner. In one embodiment, the indicia comprise a
circumferential indicator ring consisting of a plurality of
alternatingly-colored segments, the indicator ring being aligned
with the desired direction of spin of the football. The segments
are preferably selected to be of contrasting colors, and the width
of the segments is selected such that the indicator ring appears to
an observer to be a single, mixed color provided the football is
spinning about its axis at at least a predetermined frequency. In
another embodiment, the indicia comprise two or more
circumferential indicator rings at least one of which comprising a
plurality of alternatingly-colored segments. The respective segment
widths on one or more rings may be selected such that the mixed
color of one ring, and hence the spin speed of the football, can be
assessed relative to the color or mixed color of another ring.
BRIEF DESCRIPTION OF THE DRAWINGS
[0012] The present invention is best understood with reference to
the following detailed description of various embodiments of the
invention, when read in conjunction with the accompanying drawings,
in which like numerals refer to like elements, and in which:
[0013] FIG. 1 is a side view of a football having spin speed
indicia in accordance with one embodiment of the invention applied
thereon;
[0014] FIG. 2 is an end view of the football from FIG. 1;
[0015] FIG. 3 is a side view of a football having spin speed
indicia in accordance with another embodiment of the invention
applied thereon;
[0016] FIG. 4 is an end view of the football from FIG. 3;
[0017] FIG. 5 is a side view of a football having spin speed
indicia in accordance with another embodiment of the invention
applied thereon;
[0018] FIG. 6 is an end view of the football from FIG. 5;
[0019] FIG. 7 is a side view of a football having spin speed
indicia in accordance with another embodiment of the invention
applied thereon;
[0020] FIG. 8 is an end view of the football from FIG. 7;
[0021] FIG. 9 is a perspective view of a Newton's Surface in the
form of a color wheel;
[0022] FIG. 10 is a perspective view of a Newton's Surface in the
form of a cylinder;
[0023] FIG. 11 is a perspective view of a Newton's Surface in the
form of an American football;
[0024] FIG. 12 is a perspective view of a football having
circumferential indicia applied thereon;
[0025] FIG. 13 is a perspective view of a cylinder illustrating
angular and spatial relations of certain features thereof;
[0026] FIG. 14 is a perspective view of a football illustrating
angular and spatial relations of certain features thereof;
[0027] FIG. 15a is an illustration of spin speed indicia for a
football, in accordance with one embodiment of the invention;
[0028] FIG. 15b is an illustration of spin speed indicia for a
football, in accordance with an alternative embodiment of the
invention;
[0029] FIG. 15c is an illustration of spin speed indicia for a
football, in accordance with an alternative embodiment of the
invention;
[0030] FIG. 15d is an illustration of spin speed indicia for a
football, in accordance with an alternative embodiment of the
invention;
[0031] FIG. 15e is an illustration of spin speed indicia for a
football, in accordance with an alternative embodiment of the
invention; and
[0032] FIG. 16 is an illustration of spin speed indicia for a
football, in accordance with an alternative embodiment of the
invention.
DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS OF THE INVENTION
[0033] In the disclosure that follows, in the interest of clarity,
not all features of actual implementations are described. It will
of course be appreciated that in the development of any such actual
implementation, as in any such project, numerous engineering and
technical decisions must be made to achieve the developers'
specific goals and subgoals (e.g., compliance with system,
technical, and practical constraints), which will vary from one
implementation to another. Moreover, attention will necessarily be
paid to proper design and engineering practices for the environment
in question. It will be appreciated that such development efforts
could be complex and time-consuming, outside the knowledge base of
typical laymen, but would nevertheless be a routine undertaking for
those of ordinary skill in the relevant fields.
[0034] Referring to FIGS. 1 and 2 a first embodiment of the current
invention is described. In FIGS. 1 and 2, a football 1 is depicted
with respect to a longitudinal axis pp'. Football 1 has two ends
201 and 202, and pp' axis passes through the ends 201 and 202.
[0035] In accordance with one embodiment of the invention, football
1 has spin speed indicia in the form of a circumferential indicator
ring 2 applied on its surface. In the presently disclosed
embodiment, indicator ring 2 comprises alternatingly-colored
segments of a first color 3 (3a-3e) and a second color 4 (4a-4e)
substantially aligned with the desired spin direction of football 1
about its long axis. Each segment 3 is placed between two segments
4, and each segment 4 is placed between two segments 3.
[0036] In accordance with one embodiment, segments 3 and segments 4
have the same width, W. Preferably, width W is chosen so that there
are an equal number of segments 3 and segments 4. In one exemplary
embodiment, segments 3 are yellow and segments 4 are blue. In one
embodiment, five segments of each color are provided (i.e., N=5).
FIG. 2 depicts football 1 in the direction of pp' axis. It is to be
noted that in FIGS. 1 and 2, W is measured along the surface of the
ball.
[0037] When football 1 is thrown with a spin speed of R (measured
in rotations per second or RPS) about axis pp', an observer sees
segments 3 and segments 4 moving consecutively with frequency F,
measured in Hertz, where
F=R*N (Hertz, Hz)
[0038] The time it takes for one composite segment consisting of
one segment 3 and one segment 4 to replace the next such composite
segment in the progression is 1/F seconds.
[0039] If frequency F is large enough, then segments 3 and segments
4 together appear to an observer to be a single, mixed color. As
would be known to persons having ordinary skill in the art, the
resulting color is referred to as the mixed color of the two
respective original colors.
[0040] In a system producing a repetitive, periodic pattern of
colors, the smallest frequency that generates a mixed color is
called the mixing frequency, Fm, of the system. In the prior art,
the mixing frequency is sometimes called the critical flicker
frequency. Herein, the term mixing spin speed Rm (RPS) of a ring is
used to denote the rotational spin speed R corresponding to the
mixing frequency Fm. Therefore, the mixing spin speed Rm and the
mixing frequency Fm are related as follows:
Fm=Rm*N (Hz)
where N is the equal number of segments of each color comprising
the indicator ring (for example, N=5).
[0041] As would be appreciated by those of ordinary skill in the
art, the concept of a "mixing frequency" implicates the more
general concept of a so-called Newton's surface. A Newton's surface
traditionally refers to any surface having a rotational axis and
having a pattern of alternating first and second colors applied
thereon. When the surface is rotated about its axis, the colors on
a Newton's surface appear blended or mixed when the surface is
rotating at at least a certain speed; the higher the rotational
speed the more complete the blending of the colors. Nevertheless,
for any given average mean brightness, there is a rotational speed,
ms, measured in revolutions per second (RPS), above which no more
blending occurs. This speed is called the mixing spin speed (RPS),
ms, of the colored pattern on the surface.
[0042] At the mixing spin speed, all points in a plane
perpendicular to the rotational axis at a distance x from the
origin of the rotational axis, and at a distance, d, from the
rotating axis appear to be the same, single color. This color is
called the mixed color, mc, of the colored surface at distance (x,
d).
[0043] More specifically, a Newton's wheel, a Newton's cylinder,
and a Newton's Ball are referred to Newton's surfaces when the
surface is a circle or a wheel, a cylinder, and a football,
respectively. FIGS. 9-11 depict a Newton's wheel 35, a Newton's
cylinder 31, and a Newton's ball 33, respectively. Each surface has
an axis of rotation pp'. The origin of the axis pp' is denoted by
O.
[0044] At a distance (x,d), the mixing colors of the three Newton's
surfaces: Wheel, Cylinder, and Ball are denoted by mc1, mc2, and
mc3, respectively.
[0045] Referring to FIG. 10 a Newton's cylinder 31 is shown
comprising end points 216 and 217, and an indicator ring 49. FIG.
10 also depicts axis of rotation pp' having origin O, and a point
218 at distance (x, d), where x is measured from O to the plane
perpendicular to pp' and passing through the point 218, and d is
measured from point 218 to the point of intersection of pp' and the
said perpendicular plane. Ring 49 has color segments 41 and 42. The
colored segments 41 and 42 have width W. Finally the perpendicular
plane splits ring 49 into two identical rings.
[0046] Referring to FIG. 11 a Newton's ball 33 is shown comprising
end points 213 and 214, and a ring 50. FIG. 11 also depicts axis of
rotation pp' having origin O, and a point 215 at distance (x, d),
where x is measured from O to the plane perpendicular to pp' and
passing through the point 215, and d is measured from point 215 to
the point of intersection of pp' and the said perpendicular plane.
Ring 50 has color segments 43 and 44.
[0047] The width of the ring 50, referred to as WR, is defined as
follows: consider two planes perpendicular to pp'--the first plane
touching ring 50 on its side closer to the end 213, and the second
plane touching the ring 50 on its side farther away from the end
213. Now the distance between these two planes is defined to be the
width WR of ring 50. Note that WR is not related to W, width of the
color segments.
[0048] Consider further a third plane perpendicular to pp' and
between the first two planes and equidistant to both planes. The
intersection of the third plane with ring 50 is shown using a is
dashed circle in FIG. 11. This circle is called "the center line"
of the ring 50.
[0049] For a Newton's ball the width, W, of the colored segments 43
and 44 is defined as for FIG. 11. Note that in this case, W is
measured along the dashed circle.
[0050] Referring to FIG. 9, a Newton's circle or wheel 35 is shown
comprising pizza slice shape, color segments 32 and 40. FIG. 9 also
depicts axis of rotation pp' having origin O, and a point 221 at
distance (x, d), where x is measured from O to the plane
perpendicular to pp' and passing through the point 221. Since the
perpendicular plane to pp' that passes the point 221 also passed O,
then x=0. And d is measured from point 221 to the point of
intersection of pp' and the perpendicular plane.
[0051] Consider Newton's ball 33 of FIG. 11. Ball 33 has ring 50
comprising alternating segments 43 having color cl and segments 44
having color c2. The number of c1 (or c2) segments is denoted by N,
(N=4).
[0052] Ring 50 is placed on the ball such that the points on its
center line are at distance (x, d). Another way of saying this is
that a plane perpendicular to pp' at the distance x, with respect
to the origin O, would cut ring 50 through the center line of ring
50.
[0053] It is to be noted that: [0054] A. For Newton's ball 33,
parameter d is a function of x. When x is such that the
perpendicular plane is closer to ends 213 or 214 of the football,
then d is small, but when x is such that the plane is closer to the
middle of the football, then d is large. [0055] B. Referring to
FIG. 11, define the circumference of ring 50 to denote the length
of its center line. Therefore, circumference of ring 50=2*N*W,
where W is measured along the center line. Now if W and N are
fixed, then the circumference of the ring is fixed and therefore,
it can only be placed fully on a location on the football having
the same circumference. Hence, either zero, one or two places on
the football can accommodate the ring 50. More specifically, if the
circumference of ring 50 is less than the circumference of the ball
at its middle, then ring 50 can be placed on either side of the
middle of ball 33 at a unique location having the same
circumference. But if the circumference of ring 50 is greater than
the circumference of the ball at its middle, then ring 50 can't be
placed on ball 33. Finally if the circumference of ring 50 is
exactly equal to the circumference of the ball at its middle, then
ring 50 can be placed at the middle of ball 33 and at no other
location. [0056] C. On the other hand, if the size of the football
is fixed, and assuming a distance (x,d), then the parameter W,
width of the segments, becomes a function of N. For a fixed
distance (x,d), Wi is denoted by the width of color segments of a
ring similar to ring 50, when N=i, i=1,2, and so on. [0057] D.
Therefore, referring to FIG. 12, for two rings 101 and 102 on ball
34, where ring 101 has parameter N1 and it is at distance (x1,d1)
while ring 102 has parameter N2 and it is at distance (x2,d2), if
d1<d2 as shown in FIG. 12, then if N1=N2, then W1=W(ring
101)<W(ring 102)=W2.
[0058] If Newton's ball 33 of FIG. 11 is rotated faster and faster
about its axis pp', after the rotation spin speed exceeds a
threshold, ms RPS, the colors of the ring 50 will appear to be a
single mixed color, mc. The parameters, ms and mc, are the mixing
spin speed and the mixed color of the ball 33, respectively,
related to the colors on ring 50.
[0059] Ring 50 of FIG. 11 has characterization:
[0060] [{c1 c2}, c, s}], if c=mc, and s=ms.
[0061] In addition, for every spin speed, s, define a parameter f,
f=s*N, where N is as before. The parameter f is measured in Hertz,
Hz, and it refers to the number of times a pattern is repeated in
one second with respect to a stationary point close to the surface
of a Newton's surface.
[0062] For ring 50 of FIG. 11, and for a given average brightness,
one can obtain mc and ms with at least two methods:
[0063] Method 1) Use a Newton's ball. Increase the spin speed until
the colors of ring 50 appear fully mixed. The mixing spin
speed=ms=the minimum speed when the colors of ring 50 appear fully
mixed, and the mixing color=mc=the color of ring 50 when its colors
appear fully mixed.
[0064] Method 2) Human eye perception has been studied extensively.
For many average brightness levels and colors, eye response is
measured with respect to (a) frequency, f; (b) spatial frequency
fs; or (c) both. In general, for a stationary picture, the spatial
frequency, fs, is the number of times a repeated pattern falls
inside one degree of the viewing angle. More specifically,
referring to FIG. 11, assume the observer is at a distance D from
the Newton's ball looking at the ball in a direction consistent
with FIG. 11 illustration. Also assume ball 33 is stationary, i.e.,
it is not turning, then fs=the number segments of the same color
that the observer sees in one degree angle.
[0065] The measured eye responses give the so called flicker
frequencies, fk, and fks. With respect to frequency, f, color
mixing occurs at the flicker frequency, fk. And with respect to fs,
when the ring is not rotating, the repeated patterns appear uniform
and mixed at the flicker frequency, fks.
[0066] Those having ordinary skill in the art will appreciate that
there is a dependency between flicker frequencies, fk and fks. In
general, the flicker frequency, fk, drops when spatial frequency is
increased.
[0067] One way to increase spatial frequency is to place more
segments on a given ring, i.e. smaller W, or equivalently larger N.
Another way to increase spatial frequency is to view the same ring
from a farther distance--larger D. Yet another way to increase, fs,
for a given ring is to keep N fixed but reduce W, then to place the
ring closer to one of the ends of the ball.
[0068] Clearly, for a given averaged mean brightness and a ring, a
Newton's surface can be used to characterize the dependency between
the flicker frequency, fk, (and therefore, ms) and the spatial
frequency. One can place the surface at different distances and
measure the mixing spin speed (or the mixing frequency).
[0069] Fortunately, and in accordance with one aspect of the
invention, for ms belonging to a wide range of values, and for mean
brightness values corresponding to many outdoor, day conditions,
the flicker frequency, fk, of a ring for a football does not vary
significantly for observers at a few yards to a few tens of yards
from the ring. This is easily verified with the help a Newton's
surface.
[0070] It is to be noted that: [0071] (1) in most indoor or dark
outdoor conditions when average mean brightness is low, the flicker
frequency, fk, is more sensitive to the spatial frequency, fs,
variations. [0072] (2) the flicker frequency, fk, is more likely to
be significantly affected by the spatial frequency, fs, when for a
fixed N, a ring is placed too close to one of the ends of the ball
since as discussed earlier, W becomes small.
[0073] Now, if the flicker frequency, fk, is known, then we can
obtain the mixing spin speed from the equation below.
ms=fk/N.
[0074] In method (2) above it was shown that:
Given an average brightness and an N, first flicker frequency, fk,
can be derived. Then ms can be derived using ms=fk/N. The parameter
mc can be obtained using a Newton;s surface.
[0075] The following shows how to obtain N for ring 50 of FIG. 11
given the following information: [0076] a) average brightness,
[0077] b) flicker frequency, fk, and [0078] c) a desired mixing
spin speed, ms. To obtain N, use
[0078] N=fk/ms.
[0079] If N is not close to an integer, one of the following
solutions can be applied.
Solution (1): Approximate N by ceiling(N), where ceiling(N)=the
smallest integer larger than or equal to N (Note that N=fk/ms).
This selection for N would insure that when the spin speed reaches
ms, the frequency, f, f=N*s, (s=ms) is larger than the flicker
frequency. This implies that the colors are appearing mixing.
Solution (2): Approximate N by floor(N), where floor(N)=the largest
integer smaller than or equal to N (Again note that N=fk/ms).
Divide the circumference of the ring by N=fk/ms.
[0080] Let b=(circumference of the ring)/(fk/ms).
[0081] Choose floor(N) segments with color c1 and width W=b, and
choose floor(N) segments with color c2 and width W=b, then place
them on the ring alternatively.
[0082] Cover the remaining portion of the ring by color mc. Instead
of coloring the remaining portion mc, it can be covered with
shorter width color segments having mixed color equal to mc and
having mixed speed slower than ms. To the same effect, the
remaining portion of the ring can be covered as described below
with reference to FIGS. 5 and 6. Solution (2) insures that the
color mixing happen at spin speeds close to ms but it forgoes the
simple periodic color pattern of the ring.
Solution (3): Choose colors, c1 and c2, which have a different
flicker frequency, fk, for the given average brightness. More
specifically, choose colors c1 and c2 whose flicker frequency
produces an N which is close to an integer.
[0083] Any ring of more than one color situated at a distance (x,
d) on a Newton's surface will produce a mixed color at some spin
speed. The mixing spin speed, ms, and the mixing color, mc, can be
found with the help of the Newton's surface.
[0084] For example, the colored segments on a ring can have segment
shape as before or they can have more complex shapes as suggested
below.
[0085] For the sake of the present disclosure, first consider
segments for a ring on a Newton's cylinder, and then map the
segments to a ring on a Newton's ball. To this end, two functions,
function 1 and function 2, are defined that can map segments from a
Newton's cylinder to a Newton's ball.
Function 1: Referring to FIG. 13, a cylinder 55 is illustrated with
respect to an xyz coordinate system. Cylinder 55 intersects the
x-axis at (a, 0, 0) coordinate. Cylinder 55 intersects the y-axis
at (0, a, 0) coordinate.
[0086] Cylinder 55 is characterized by the following equation.
(x.sup.2+y.sup.2)/a.sup.2=1
[0087] Referring to FIG. 14, a ball 56 is illustrated with respect
to an xyz coordinate system. Ball 56 intersects the x-axis at (a,
0, 0) coordinate. Ball 56 intersects the y-axis at (0, a, 0)
coordinate. Ball 56 intersects the z-axis at (0, 0, b)
coordinate.
[0088] Cylinder 55 is characterized by the following equation.
(x.sup.2+y.sup.2)/a.sup.2+z.sup.2/b.sup.2=1
[0089] Given a point A on cylinder 55, the following parameters are
defined:
[0090] Az=distance of point A to xy plane
[0091] t=the angle between x-axis and the line passing through the
origin, R, of the xyz coordinates, and the point A', which is the
projection of A onto xy plane. The angle is measured
counterclockwise in the xy plane looking from positive z half of
the space.
[0092] Function 1 maps point A to a point B on ball 56, having
coordinates (Bx, By, Bz) where
Bx=sqrt(a.sup.2.times.(1-Az.sup.2/b.sup.2)).times.cos(t)
By=sqrt(a.sup.2.times.(1-Az.sup.2/b.sup.2)).times.sin(t)
Bz=Az
[0093] Function 1 maps a segment, Q1, on cylinder 55 to a segment,
Q2, on ball 56 by mapping the points on Q1 to points on ball 56. Of
course in general not every point need to be mapped and only enough
points to enable a close approximation of Q2.
[0094] Function 2: To describe function 2, use FIGS. 13 and 14 as
before except do not assume the following characterization of ball
56.
(x.sup.2+y.sup.2)/a.sup.2+z.sup.2/b.sup.2=1
[0095] Given a point A on cylinder 55, we define parameters Az and
t as before.
[0096] Function 2 maps point A to a point B on ball 56, where B has
coordinates (Bx, By, Bz) and
Bx=cos(t).times.Circumference(Az)/(2.times.pi)
By=sin(t).times.Circumference(Az)/(2.times.pi)
Bz=Az
where Circumference(Az) is the circumference of ball 56 measured at
z=Az.
[0097] Function 2 maps any segment, Q1, on cylinder 55 to a
segment, Q2, on ball 56.
Below color segments are described for rings on Newton's cylinder.
Each segment generates a segment on Newton's ball using function 1
or function 2.
Briefly,
[0098] (1) Color segments can be segments or closed segments.
[0099] (2) Segments can be identical or not. 21 More specifically,
color segments can belong to any class or subclass below:
[0100] A. Identical segments [0101] a. of rectangular shape, as
shown in FIG. 15a. [0102] b. of parallelogram shape, as shown in
FIG. 15b. [0103] c. with curved boundaries, as shown in FIG.
15c.
[0104] B. Non-identical segments [0105] a. shown in FIG. 15d as
having two colors: c1 and c2.
[0106] C. Identical closed segments [0107] a. of triangular shape,
as shown in FIG. 15e.
[0108] Referring to FIG. 15e, a band, 300, comprising triangles of
two colors is shown. If band 300 is cut into narrower bands 301,
302 and 303, then bands 301 and 303 would have higher mixed spin
speeds than ring 302. Ring 300 will appear having uniform color
when all its narrow bands reach their mixed spin speeds. In another
words, when spin speed reaches the mixed spin speed of a sub-band
of ring 300 having the largest spin speed, then ring 300 will
appear having mixed colors. However, since different bands might
have different mixed spin speeds, the sub-bands with lower mixed
spin speed will appear mixed before the sub-bands with larger mixed
spin speeds. These variations provide an advantage in that the
lower mixed spin speed bands would provide "a reference color" for
the larger mixed spin speed bands--similar to ring 5 and ring 6
(described below), respectively.
[0109] D. More generally any pattern that produces a mixed color at
a predetermined spin speed can be depicted on the ring.
[0110] In one embodiment, the two colors of an indicator ring 2
from FIG. 1 are yellow and blue, and the mixed color is green. The
mixing frequency ranges from 20 to 80 Hz, depending on the mean
brightness conditions--the brighter the conditions, the higher the
critical frequency. See, Yao Wang et al., "Video Processing and
Communications," ISBN No. ______, Wiley, 2005. If average mean
brightness requiring a critical flicker frequency of 50 Hz is
assumed, then a spin speed,
R=Fm/N=50 Hz/5=10 (RPS)
will produce the mixed color, green, to the observer. Therefore,
Rm=10 RPS under the brightness assumption above.
[0111] In accordance with one aspect of the invention, football 1
helps players practice achieving a desired spin speed of a thrown
ball. If segments 3 and segments 4 do not produce the appearance of
the mixed color, then players would know that additional spin speed
is required. On the other hand, if the mixed color is produced,
then players are provided positive feedback regarding their spin
speed skill.
[0112] In addition, players working on their spin speed can measure
their progress when they produce the mixed color if initially
failing to do so.
[0113] The following notation summarizes some of the color mixing
parameters of football 1.
[0114] A ring similar to ring 2 in FIGS. 1 and 2 has characteristic
I given by
I(ring)=[{c1,c2}, m, s]
[0115] if the two colors are c1 and c2, the mixed color is m, and
the mixing spin speed is s (RPS), for a given brightness.
[0116] Using this convention, an indicator ring 2 in an exemplary
embodiment may have the following characteristic:
I(ring 2)=[{yellow, blue}, green, 10 RPS]
assuming an average mean brightness requiring a critical flicker
frequency of 50 Hz.
[0117] Those of ordinary skill in the art having the benefit of the
present disclosure will appreciate that in order to produce a
slower mixing spin speed for a given football 1, the characteristic
N can be increased. If N is increased from five to ten, for
example, then the mixing spin speed Rm from the above example
becomes
Rm=Fm/N=50/10=5 RPS.
[0118] This is one-half of the initial mixing spin speed (10 RPS).
Lower mixing spin speeds might often be desirable. For example, a
quarterback might plan to acquire the skill of spinning 10 RPS by
practicing with five footballs 1a, 1b, 1c, 1d, and 1e having mixing
spin speeds 6 RPS, 7 RPS, 8 RPS, 9 RPS, and 10 RPS, respectively.
He would start with football 1a and then switch to football 1b when
he feels comfortable spinning 6 RPS. Then he would switch to
football 1c and so on as he masters each ball.
[0119] Also for a child, football player a goal of achieving a
small mixing spin speed obviously is more practical than a large
mixing spin speed.
[0120] Turning now to FIGS. 3 and 4, there is shown a football 10
in accordance with an alternative embodiment of the invention. In
particular, FIG. 3 illustrates a football 10 having ends 203 and
204, and having an indicator ring 2 substantially the same as in
the embodiment of FIGS. 1 and 2 (and hence retaining the same
reference numeral in FIGS. 3 and 4). In addition to its ring 2,
football 10 in FIGS. 3 and 4 has a second indicator ring 5. Ring 5
is preferably of a single color, and more specifically it is
preferably the same color as the mixed color of ring 2.
[0121] Thus, for football 10, the characteristic I is given by:
I(ring 2)=[{yellow, blue}, green, 10 RPS]
where Color(ring 5)=green=the mixed color of ring 2. (Herein, the
notation Color(X) is used to to denote the color of entity X.)
[0122] As previously described, assuming an average mean
brightness, ring 2 of football 10 will appear to an observer to
have its mixed color if football 10 has a spin speed of R=10 RPS.
Therefore, in the embodiment of FIGS. 3 and 4, rings 2 and 5 would
appear having the same color when R=10 RPS. Thus ring 5 provides a
"reference color for the mixed color of ring 2 since Color(ring
5)=the mixed color m of ring 2.
[0123] Similar to football 1, football 10 helps players practice
achieving a desired spin speed. However, for football 10 it is
believed to be easier to recognize the mixing of colors of ring 2
having it juxtaposed with ring 5, which has the mixed color of ring
2.
[0124] Those of ordinary skill will appreciate that increasing of
the spin speed beyond the mixing spin speed would not change the
appearance of ring 2. That is, once colors appear mixed, any faster
spin speed does not change that appearance.
[0125] Turning now to FIGS. 5 and 6, there is shown a football 20
in accordance with another 11 embodiment of the invention. FIG. 5
illustrates a football 20 having ends 205 and 206, and having ring
2 of the first embodiment. In addition to its ring 2, football 20
has a second ring 6. Ring 6 is similar to ring 2 except that the
width, w, of its yellow and blue segments are smaller than W, the
width of segments 3 and 4 of ring 2.
[0126] Therefore, football 20 has the following characteristics
I(ring 2)=[{yellow, blue}, green, 10 RPS], and
I(ring 6)=[{yellow, blue}, green, r RPS],
where r is much smaller than 10 RPS. For ring 6 of FIG. 5, r=10/3
RPS.
[0127] Again assuming an average mean brightness, ring 2 will
appear to an observer as the mixed color if football 20 is spun
with spin speed R=10 RPS. However ring 6 will appear as the mixed
color at lower spin speeds, even at speed spin as small as r=10/3
revolution per second. Therefore, for spin speeds of interest--10
RPS for football 20--ring 6 will appear green and it can provide
the reference color for the mixed color of ring 2.
[0128] Hence ring 6 in FIGS. 5 and 6 and ring 5 in FIGS. 3 and 4,
respectively, perform the same function; namely that of they
providing a reference color for the mixed color of ring 2. Note
that r can easily be made smaller than 10/3 by reducing w.
[0129] An advantage of ring 6 in FIGS. 5 and 6 over ring 5 in FIGS.
3 and 4 is that ring 6 provides a more robust reference for the
mixed color. That is, in FIGS. 3 and 4, if the mixed color (e.g.
green) used in ring 5 differs from the "true" mixed color due to
wear and tear or other reasons, then the observer will not see
closely matched colors in rings 2 and 5.
[0130] FIGS. 7 and 8 illustrate still another embodiment of the
present invention. FIG. 7 depicts a football 30 having a ring 2
substantially the same as ring 2 in the above-described
embodiments. In addition to ring 2, football 30 has a second ring
9. Ring 9 comprises segments: 7a-7d, having a third color, c3, and
segments: 8a-8d having a fourth color, c4. Each segment 7 is placed
between two segments 8, and each segment 8 is placed between two
segments 7. Segments 7 and 8 both have width W2, as shown in FIGS.
7 and 8. Again W2 is measured along the surface of football 30.
Football 30 has ends 207 and 208.
[0131] Football 30 thus has the following characteristics:
I(ring 2)=[{yellow, blue}, green, 10 RPS], and
I(ring 9)=[{c3, c4}, m2, s2 RPS].
[0132] Ring 9 in the embodiment of FIGS. 7 and 8 differs from ring
5 in FIGS. 3 and 4 and ring 6 of FIGS. 5 and 6. In particular, ring
5 and ring 6 provide "a reference color" for the mixed color of
ring 2. But ring 9 in the forth embodiment provides advantages
explained below.
[0133] Again assuming an average mean brightness, ring 2 will
appear to an observer as a mixed color if football 30 is spun with
spin speed R=10 RPS.
[0134] There are eight variations to ring 9. These variations and
their advantages are given in Table 1, where c1=yellow, c2=blue,
m1=green, and s1=10 RPS
TABLE-US-00001 TABLE 1 Variation Parameters of ring 9 Advantage
provided by ring 9 1 c3 = c1 more confirmation c4 = c2 m2 = m1 s2 =
s1 2 c3 = c1 estimate progress c4 = c2 m2 = m1 s2 .noteq. s1 3 c3 =
c1 more confirmation c4 = c2 m2 .noteq. m1 s2 = s1 4 c3 = c1
estimate progress c4 = c2 m2 .noteq. m1 s2 .noteq. s1 5 c3 .noteq.
c1 more confirmation c4 .noteq. c2 m2 = m1 s2 = s1 6 c3 .noteq. c1
estimate progress c4 .noteq. c2 m2 = m1 s2 .noteq. s1 7 c3 .noteq.
c1 more confirmation c4 .noteq. c2 m2 .noteq. m1 s2 = s1 8 c3
.noteq. c1 estimate progress c4 .noteq. c2 m2 .noteq. m1 s2 .noteq.
s1
[0135] Variations 1-8 are explained below. Although N=5 for ring 2,
and N=4 for ring 9, these numbers are chosen for illustration
purposes and they are not assumed to be fixed.
Variation 1:
[0136] c3=c1; c4=c2; m2=m1; s2=s1; [0137] The two rings make it
easier to observe the mixed color hence they make it easier to
validate achieving the mixing speed. This variation can be realized
with
[0137] N(ring 9)=N(ring 2).
Variation 2:
[0138] c3=c1; c4=c2; m2=m1; s2.noteq.s1, s2>s1; [0139] The two
rings have different mixing spin speeds; ring 2 reaches its mixing
spin speed sooner than ring 9. This variation provides a method to
evaluate one's progress in spinning football 30. If ring 2 appears
color m1 but ring 9 colors, c3 and c4, do not appear mixed, then
this indicates to one the need to improve the spin. Progress is
confirmed when ring 9 appears color m2. Note in this variation
m2=m1. This variation can be realized with
[0139] N(ring 9)<N(ring 2).
Variation 3:
[0140] c3=c1; c4=c2; m2.noteq.m1; s2=s1; [0141] The two rings have
different mixed colors but the same mixing spin speed. This
variation is similar to variation 1; the two rings make it easier
to confirm that the mixing speed is achieved. This variation can be
realized when FOR THE same ring (ring 2 or ring 9),
[0141] W(one color segment).noteq.W(the other color segment).
Variation 4:
[0142] c3=c1; c4=c2; m2.noteq.m1; s2.noteq.s1, s2>s1; [0143] The
two rings have different mixing spin speeds and different mixed
colors; ring 2 reaches its mixing speed sooner than ring 9. This
variation works similar to variation 2. If ring 2 appears m1 but
ring 9 colors, c3 and c4, do not appear mixed, then this indicates
the need to improve the spin. Progress is confirmed when ring 9
appears as color m2. Note in this variation that m2.noteq.m1. This
variation can be realized when for the same ring,
W(c3).noteq.W(c4).
Variation 5:
[0143] [0144] c3.noteq.c1; c4.noteq.c2; m2=m1; s2=s1; [0145] The
two rings have different colors but the same mixed color and the
same mixing speed. The advantage of variation 5 is very similar to
that of variations 1 and 3. The two rings make it easier to observe
the mixed color hence they make it easier to validate achieving the
mixing speed.
Variation 6:
[0145] [0146] c3.noteq.c1; c4.noteq.c2; m2=m1; s2.noteq.s1,
s2>s1; [0147] The two rings have different colors and different
mixing speeds; again, ring 2 reaches its mixing speed sooner than
ring 9. This variation provides another method to evaluate one's
progress in spinning the football 30. If ring 2 appears color ml
but for ring 9 colors, c3 and c4, do not appear mixed, then this
indicates the need to improve the spin. Progress is confirmed when
ring 9 appears color m2. Note in this variation m2=m1. This
variation can be realized with N(ring9)<N(ring2).
Variation 7:
[0147] [0148] (c1 c2).noteq.(c3 c4); m2.noteq.m1; s2=s1; [0149] The
two rings 2 and 9 do not have the same colors. That is, at least
one of their colors differ. Also they have different mixed colors
but the same mixing spin speed. This variation is similar to
variations 1, 3 and 5; the two rings make it easier to confirm that
the mixing speed is achieved.
Variation 8:
[0149] [0150] (c1 c2).noteq.(c3 c4); m2.noteq.m1; s2.noteq.s1,
s2>s1; [0151] The two rings 2 and 9 do not have the same colors.
That is, at least one of their colors differ. Also 8 they have
different mixing spin speeds and different mixed colors; ring 2
reaches its mixing speed sooner than ring 9. This variation works
similar to variations 2, 4 and 6. If ring 2 appears m1 but ring 9
colors, c3 and c4, do not appear mixed, then this indicates the
need to improve the spin. Progress is confirmed when ring 9 appears
color m2. Note in this variation m2.noteq.m1. This variation can be
realized with N(ring9)<N(ring2).
[0152] In the description of the embodiments, width W was measured
along one side of the ring. Those of ordinary skill having the
benefit of the present disclosure will recognize that W may be
measured according to various measuring conventions. Nonetheless,
given a ring, its width, WR, and its location on the ball, as long
as it is stated where W is measured there will be no ambiguity.
[0153] Although the embodiments described herein involve footballs
with one or two rings, other variations or generalizations have
been conceived, including: (I) a football with more than two rings;
(II) a football with more than two rings where at least three of
the rings have distinct mixed colors; (III) a football with more
than two rings where at least three of rings: ring 1, ring 2, and
ring 3 have distinct mixing spin speeds s1, s2, and s3, where
s1<s2<s3. If ring 1 colors appear mixed but ring 2 and ring 3
colors do not appear mixed, then this indicates that spin speed s1
is achieved bt we need to practice to achieve s2 and s3. If ring 1
and ring 2 colors appear mixed but ring 3 colors do not, then this
indicates further progress but with a need to practice to achieve
s3. When all rings appear in their mixed color then the goal is
achieved. Therefore, the combination of ring 1, ring 2, and ring 3
enable a more graduated method to measure progress than two
rings.
[0154] It is also contemplated that football 30 of the embodiment
of FIG. 7 may support an additional "reference ring" similar to
ring 5 of the embodiment of FIG. 3. The added ring can provide "a
reference color" corresponding to the mixed color of one of the
rings: ring 2 or ring 9 of the embodiment of FIG. 7.
[0155] Furthermore, football 30 of the embodiment of FIG. 7 may
support an additional "reference ring" similar to ring 6 of the
embodiment of FIG. 5. The added ring can provide "a reference
color" corresponding to the mixed color of one of the rings: ring 2
or ring 9 of the embodiment of FIG. 7.
[0156] In one embodiment of the invention, it is contemplated that
a set, S, containing a predetermined spin speeds is generated. For
example S={s1=2 RPS, s2=3 RPS, s3=4 RPS, s4=5 RPS}. The spin speeds
in S are selected to be suitable for spin training in throwing a
football. Next for each spin speed in the set S, a football
according to one of the foregoing embodiments is produced. A
trainee would start with the football with the ring having a mixing
spin speed, s1. He would practice until he achieves spins causing
the mixed color on the ring on the football. The trainee will then
move up to the football with the ring having the mixing spin speed,
s2, and so on.
[0157] In all given embodiments one may use a ring having three or
more color segments juxtaposed in a sequential pattern. In terms of
the nomenclature used herein, I(ring of three
colors)=[{c1,c2,c3},m,s]: For example, a ring can have three colors
c1, c2, and c3 where the colors are arranged periodically where c2
follows c1 and c3 follows c2 and then c1 follows c3 again.
[0158] As previously noted, the colored segments do not have to be
segments. FIG. 16 depicts a ring 110 having colored segments
comprising closed segments that are portions of six sided polygons.
Ring 110 has the following representation:
I(ring 110)=[{c1,c2,c3}, m, s].
[0159] For a given average mean brightness, an easy way to find the
mixing spin speed, ms, and 9 the mixed color, mc, of a ring with
more than two colors from a distance D, (D=distance between the
observer and the ring) is to use a Newton's surface as before.
[0160] Referring to FIG. 16, if one cuts a band 110 into narrower
bands 111, 112, and 113, then bands 111 and 113 would have higher
mixed spin speeds than band 112. Ring 110 will appear having
uniform color when all its narrow bands reach their mixed spin
speeds. In another words, when spin speed reaches the mixed spin
speed of a sub-band of the ring 110 having the largest spin speed,
then ring 110 will appear having mixed colors. However, since
different bands might have different mixed spin speeds, the
sub-bands with lower mixed spin speed will appear mixed before the
sub-bands with larger mixed spin speeds. Again the variations on
the mixed spin speed provide an advantage that the lower mixed spin
speed bands would provide "a reference color" for the larger mixed
spin speed bands, similar to ring 5 and ring 6 in the embodiments
of FIGS. 3 and 5, respectively.
[0161] In another alternative embodiment, colored rings can be
formed on footballs by painting the surface of the ball. They can
also be formed by adhering a colored band on the football.
[0162] Also, if the band is elastic with enough gripping, then it
can be fixed to the surface without using any adhesives. Elastic
bands provide an additional advantage in that they can be removed
and be replaced by other colored bands.
[0163] If the flicker frequency change is pronounced, due for
example to a spatial frequency change resulting from the ball
moving away from the observer, then the ball can be designed
according to the embodiment of FIG. 7 (cases 2, 4, 6, or 8). More
specifically, partition the special frequency into two sets: low
frequency and high frequency. Select a flicker frequency, fk1,
produced when spatial frequency is in low frequency set, and select
a flicker frequency, fk2, produced when spatial frequency is in
high frequency set. Now match ring 2 mixing spin speed to fk1, and
match ring 9 mixing spin speed to fk2. All we need to do now is to
observe ring 2 for color mixing when the ball is near and to
observe ring 9 when the ball is far.
[0164] When applying the use of the present invention to a group of
users having a same age, flicker frequency variations may be
assumed negligible.
[0165] From the foregoing disclosure, it should be apparent that a
method and apparatus for indicating spin speed of a thrown football
has been disclosed.
[0166] Although a specific embodiment of the invention as well as
possible variants and alternatives thereof have been described
and/or suggested herein, it is to be understood that the present
disclosure is intended to teach, suggest, and illustrate various
features and aspects of the invention, but is not intended to be
limiting with respect to the scope of the invention, as defined
exclusively in and by the claims, which follow.
[0167] Indeed, it is contemplated and to be explicitly understood
that various substitutions, alterations, and/or modifications,
including but not limited to any such implementation variants and
options as may have been specifically noted or suggested herein,
including inclusion of technological enhancements to any particular
method step or system component discovered or developed subsequent
to the date of this disclosure, may be made to the disclosed
embodiment of the invention without necessarily departing from the
technical and legal scope of the invention as defined in the
following claims.
* * * * *