Image generating apparatus, image generating method, entertainment system, and recording medium
View Patent ↗An image generating apparatus and image generating method determine the distribution state of dot-sequences on a two-dimensional screen using two-dimensional Gaussian distribution in the event that the plurality of dot-sequences randomly exist in a sphere within the three-dimensional space, an generate an image of dot-sequences on the two-dimensional screen based on the distribution state on the two-dimensional screen. The two-dimensional Gaussian distribution can be factored to the product of one-dimensional Gaussian distributions, and the one-dimensional Gaussian distributions can be generated by uniform random numbers. Thus, it is easy to create an image by projecting a plurality of dot-sequences randomly existing in a three-dimensional space onto a two-dimensional screen.
1. An apparatus for projecting a plurality of dot-sequences existing in a three-dimensional space upon a hypothetical two-dimensional screen, said apparatus comprising a graphics data generating processor operable to determine a spatial distribution state of said dot-sequences on said two-dimensional screen based on a spatial distribution state of said
dot-sequences in said three-dimensional space, given that said plurality of dot-sequences randomly exist in a domain within said three-dimensional space, and to generate an image of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said dot-sequences on said two-dimensional screen.
2. An apparatus according to claim 1 , wherein
said spatial distribution state of said dot-sequences on said two-dimensional screen is represented by a probability said spatial distribution state of said dot-sequences in said three-dimensional space is represented by a probability density function in said three-dimensional space.
3. An apparatus for projecting a plurality of dot-sequences existing in a three-dimensional space upon a hypothetical two-dimensional screen, said apparatus comprising a graphics data generating processor to
determine a spatial distribution state of said dot-sequences on said two-dimensional screen based on a spatial distribution state of said dot-sequences in said three-dimensional space, assuming that said plurality of dot-sequences randomly exist in a domain within said three-dimensional space;
determine a point of view within said three-dimensional space from which to provide a reference for projecting said dot-sequences onto said two-dimensional screen;
determine a direction of view from said point of view toward said plurality of dot-sequences; and
generate an image of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said dot-sequences on said two-dimensional screen.
4. An apparatus according to claim 3 , wherein said distribution state of said dot-sequences on said two-dimensional screen is a probability density function g(x, y) expressed as follows:
g
(
x
,
y
)
=
∫
0
∞
f
(
X
,
Y
,
Z
)
·
(
Z
/
h
)
2
ⅆ
Z
,
wherein
said three-dimensional space is defined by x, y, and z coordinate axes;
said direction of view from said point of view toward said plurality of dot-sequences is the z axis;
said two-dimensional screen is oriented along the x and y axes;
a probability density function for said plurality of dot-sequences existing within said three-dimensional space is f(X, Y, Z);
a distance from said point of view to said two-dimensional screen is h; and
a distance from said point of view to said dot-sequences existing within said three-dimensional space is Z.
5. An apparatus according to claim 3 , wherein said distribution state of said dot-sequences on said two-dimensional screen is a probability density function g(x, y) expressed as follows:
g
(
x
,
y
)
=
∫
0
∞
f
(
x
·
Z
/
h
,
y
·
Z
/
h
,
Z
)
·
(
Z
/
h
)
2
ⅆ
Z
,
wherein
said three-dimensional space is defined by x, y, and z coordinate axes;
said direction of view from said point of view toward said plurality of dot-sequences is the z axis;
said two-dimensional screen is oriented along the x and y axes;
a probability density function for said plurality of dot-sequences existing within said three-dimensional space is f(X, Y, Z);
a distance from said point of view to said two-dimensional screen is h; and
a distance from said point of view to said dot-sequences existing within said three-dimensional space is Z.
6. An apparatus according to claim 3 , wherein said plurality of dot-sequences existing within said three-dimensional space are apex coordinates of polygons which are used to draw image objects.
7. An apparatus according to claim 3 , wherein said plurality of dot-sequences existing within said three-dimensional space are coordinates for drawing at least one object from the group consisting of clouds, a water surface, and tree leaves.
8. An apparatus according to claim 3 , wherein
said spatial distribution state of said dot-sequences on said two-dimensional screen is represented by a probability density function on said two-dimensional screen, and
said spatial distribution state of said dot-sequences in said three-dimensional space is represented by a probability density function in said three-dimensional space.
9. An apparatus for projecting a plurality of dot-sequences existing in a three-dimensional space upon a hypothetical two-dimensional screen, said apparatus comprising a graphics data generating processor operable to determine a spatial distribution state of said dot-sequences on said two-dimensional screen using uniform random numbers, given that said plurality of dot-sequences randomly exist in a domain within said three-dimensional space, and to generate an image of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said dot-sequences on said two-dimensional screen.
10. An apparatus for projecting a plurality of dot-sequences existing in a three-dimensional space upon a hypothetical two-dimensional screen, said apparatus comprising a graphics data generating processor operable to determine a spatial distribution state of said dot-sequences on said two-dimensional screen based on a one-dimensional Gaussian distribution generated from uniform random numbers using the Box-Muller method, given that said plurality of dot-sequences randomly exist in a domain within said three-dimensional space, and to generate an image of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said dot-sequences on said two-dimensional screen.
11. An apparatus for projecting a plurality of dot-sequences existing in a three-dimensional space upon a hypothetical two-dimensional screen, said apparatus comprising a graphics data generating processor operable to determine a spatial distribution state of said dot-sequences on said two-dimensional screen using a two-dimensional Gaussian distribution, given that said plurality of dot-sequences randomly exist in a domain within said three-dimensional space, and to generate an image of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said dot-sequences on said two-dimensional screen.
12. An apparatus according to claim 11 , wherein said distribution state of said dot-sequences on said two-dimensional screen is a probability density function g(x, y) expressed as follows:
g ( x,y )=(1/2π)exp{− x 2 +y 2 )/2},
wherein
said three-dimensional space is defined by x, y, and z coordinate axes; and
said two-dimensional screen is oriented along the x and y axes.
13. A method for projecting a plurality of dot-sequences existing in a three-dimensional space upon a hypothetical two-dimensional screen, said method comprising:
determining a spatial distribution state of said dot-sequences in said three-dimensional space, given that said plurality of dot-sequences randomly exist in a domain within said three-dimensional space;
determining a spatial distribution state of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said plurality of dot-sequences in said three-dimensional space; and
generating an image of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said dot-sequences on said two-dimensional screen.
14. A method according to claim 13 wherein
said spatial distribution state of said dot-sequences on said two-dimensional screen is represented by a probability density function on said two-dimensional screen, and
said spatial distribution state of said dot-sequences in said three-dimensional space is represented by a probability density function in said three-dimensional space.
15. A method for projecting a plurality of dot-sequences existing in a three-dimensional space upon a hypothetical two-dimensional screen, said method comprising:
determining a spatial distribution state of said dot-sequences in said three-dimensional space, given that said plurality of dot-sequences randomly exist in a domain within said three-dimensional space;
determine a point of view within said three-dimensional space from which to provide a reference for projecting said dot-sequences onto said two-dimensional screen;
determining a direction of view from said point of view toward said plurality of dot-sequences;
determining a spatial distribution state of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said dot-sequences, in said three-dimensional space, said point of view, and said direction of view; and
generating an image of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said dot-sequences on said two-dimensional screen.
16. A method according to claim 15 , wherein said distribution state of said dot-sequences on said two-dimensional screen is a probability density function g(x, y) expressed as follows:
g
(
x
,
y
)
=
∫
0
∞
f
(
X
,
Y
,
Z
)
·
(
Z
/
h
)
2
ⅆ
Z
,
wherein
said three-dimensional space is defined by x, y, and z coordinate axes;
said direction of view from said point of view toward said plurality of dot-sequences is the z axis;
said two-dimensional screen is oriented along the x and y axes;
a probability density function for said plurality of dot-sequences existing within said three-dimensional space is f(X, Y, Z);
a distance from said point of view to said two-dimensional screen is h; and
a distance from said point of view to said dot-sequences existing within said three-dimensional space is Z.
17. A method according to claim 15 , wherein said distribution state of said dot-sequences on said two-dimensional screen is a probability density function g(x, y) expressed as follows:
g
(
x
,
y
)
=
∫
0
∞
f
(
x
·
Z
/
h
,
y
·
Z
/
h
,
Z
)
·
(
Z
/
h
)
2
ⅆ
Z
,
wherein
said three-dimensional space is defined by x, y, and z coordinate axes;
said direction of view from said point of view toward said plurality of dot-sequences is the z axis;
said two-dimensional screen is oriented along the x and y axes;
a probability density function for said plurality of dot-sequences existing within said three-dimensional space is f(X, Y, Z);
a distance from said point of view to said two-dimensional screen is h; and
a distance from said point of view to said dot-sequences existing within said three-dimensional space is Z.
18. A method according to claim 15 , wherein said plurality of dot-sequences existing within said three-dimensional space are apex coordinates of polygons which are used to draw image objects.
19. A method according to claim 16 , wherein said plurality of dot-sequences existing within said three-dimensional space are coordinates for drawing at least one object selected from the group consisting of clouds, a water surface, and tree leaves.
20. A method according to claim 15 wherein
said spatial distribution state of said dot-sequences on said two-dimensional screen is represented by a probability density function on said two-dimensional screen, and
said spatial distribution state of said dot-sequences in said three-dimensional space is represented by a probability density function in said three-dimensional space.
21. A method for projecting a plurality of dot-sequences existing in a three-dimensional space upon a hypothetical two-dimensional screen, given that said plurality of dot-sequences randomly exist in a domain within said three-dimensional space, said method comprising:
determining a spatial distribution state of said dot-sequences on said two-dimensional screen using uniform random numbers; and
generating an image of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said dot-sequences on said two-dimensional screen.
22. A method for projecting a plurality of dot-sequences existing in a three-dimensional space upon a hypothetical two-dimensional screen, given that said plurality of dot-sequences randomly exist in a domain within said three-dimensional space, said method comprising:
determining a spatial distribution state of said dot-sequences on said two-dimensional screen based on a one-dimensional Gaussian distribution generated from uniform random numbers using the Box-Muller method; and
generating an image of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said dot-sequences on said two-dimensional screen.
23. A method for projecting a plurality of dot-sequences existing in a three-dimensional space upon a hypothetical two-dimensional screen, given that said plurality of dot-sequences randomly exist in a domain within said three-dimensional space, said method comprising:
determining a spatial distribution state of said dot-sequences on said two-dimensional screen using a two-dimensional Gaussian distribution; and
generating an image of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said dot-sequences on said two-dimensional screen.
24. A method according to claim 23 , wherein said distribution state of said dot-sequences on said two-dimensional screen is a probability density function g(x, y) expressed as follows:
g ( x,y )=(1/2π)exp{−( x 2 +y 2 )/2},
wherein
said three-dimensional space is defined by x, y, and z coordinate axes; and
said two-dimensional screen is oriented along the x and y axes.
25. An entertainment system for producing images for display on a display screen, said system comprising a graphics system including a graphics data generating processor operable to perform coordinate computation on primitives of said images, and given that a plurality of dot-sequences randomly exist in a domain within a three-dimensional space, said graphics data generating processor is further operable to determine a spatial distribution state of said dot-sequences on a two-dimensional screen based on a spatial distribution state of said dot-sequences in said three-dimensional space, and to generate an image of said dot-sequences on said two-dimensional screen according to said spatial distribution state of said dot-sequences on said two-dimensional screen.
26. An entertainment system for producing images for display on a display screen, said system comprising a graphics system including a graphics data generating processor operable to perform coordinate computation on primitives of said images, and given that a plurality of dot-sequences randomly exist in a domain within a three-dimensional space, said graphics data generating processor is further operable to: determine a spatial distribution state of said dot-sequences on a two-dimensional screen based on a spatial distribution state of said dot-sequences in said three-dimensional space, a point of view within said three-dimensional space from which to provide a reference for projecting said dot-sequences onto said two-dimensional screen, and a direction of view from said point of view toward said plurality of dot-sequences; and to generate an image of said dot-sequences on said two-dimensional screen according to said spatial distribution state of said dot-sequences on said two-dimensional screen.
27. An entertainment system for producing images for display on a display screen, said system comprising a graphics system including a graphics data generating processor operable to perform coordinate computation on primitives of said images, and given that a plurality of dot-sequences randomly exist in a domain within a three-dimensional space, said graphics data generating processor is further operable to determine a spatial distribution state of said dot-sequences on a two-dimensional screen using a two-dimensional Gaussian distribution, and to generate an image of said dot-sequences on said two-dimensional screen according to said spatial distribution state of said dot-sequences on said two-dimensional screen.
28. A storage medium upon which a program is stored, said program being operable to cause a processing system to project a plurality of dot-sequences existing in a three-dimensional space upon a hypothetical two-dimensional screen by performing steps comprising:
determining a spatial distribution state of said dot-sequences in said three-dimensional space, given that said plurality of dot-sequences randomly exist in a domain within said three-dimensional space;
determining a spatial distribution state of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said plurality of dot-sequences in said three-dimensional space; and
generating an image of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said dot-sequences on said two-dimensional screen.
29. A storage medium according to claim 28 , wherein
said spatial distribution state of said dot-sequences on said two-dimensional screen is represented by a probability density function on said two-dimensional screen, and
said spatial distribution state of said dot-sequences in said three-dimensional space is represented by a probability density function in said three-dimensional space.
30. A storage medium upon which a program is stored, said program being operable to cause a processing system to project a plurality of dot-sequences existing in a three-dimensional space upon a hypothetical two-dimensional screen by performing steps comprising:
determining a spatial distribution state of said dot-sequences in said three-dimensional space, given that said plurality of dot-sequences randomly exist in a domain within said three-dimensional space;
determining a point of view within said three-dimensional space from which to provide a reference for projecting said dot-sequences onto said two-dimensional screen;
determining a direction of view from said point of view toward said plurality of dot-sequences;
determining a spatial distribution state of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said dot-sequences in said three-dimensional space, said point of view, and said direction of view; and
generating an image of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said dot-sequences on said two-dimensional screen.
31. A storage medium according to claim 25 , wherein said distribution state of said dot-sequences on said two-dimensional screen is a probability density function g(x, y) expressed as follows:
g
(
x
,
y
)
=
∫
0
∞
f
(
X
,
Y
,
Z
)
·
(
Z
/
h
)
2
ⅆ
Z
,
wherein
said three-dimensional space is defined by x, y, and z coordinate axes;
said direction of view from said point of view toward said plurality of dot-sequences is the z axis;
said two-dimensional screen is oriented along the x and y axes;
a probability density function for said plurality of dot-sequences existing within said three-dimensional space is f(X, Y, Z);
a distance from said point of view to said two-dimensional screen is h; and
a distance from said point of view to said dot-sequences existing within said three-dimensional space is Z.
32. A storage medium according to claim 30 , wherein said distribution state of said dot-sequences on said two-dimensional screen is a probability density function g(x, y) expressed as follows:
g
(
x
,
y
)
=
∫
0
∞
f
(
x
·
Z
/
h
,
y
·
Z
/
h
,
Z
)
·
(
Z
/
h
)
2
ⅆ
Z
,
wherein
said three-dimensional space is defined by x, y, and z coordinate axes;
said direction of view from said point of view toward said plurality of dot-sequences is the z axis;
said two-dimensional screen is oriented along the x and y axes;
a probability density function for said plurality of dot-sequences existing within said three-dimensional space is f(X, Y, Z);
a distance from said point of view to said two-dimensional screen is h; and
a distance from said point of view to said dot-sequences existing within said three-dimensional space is Z.
33. A storage medium according to claim 30 , wherein said plurality of dot-sequences existing within said three-dimensional space are apex coordinates of polygons which are used to draw image objects.
34. A storage medium according to claim 30 , wherein said plurality of dot-sequences existing within said three-dimensional space are coordinates for drawing at least one object selected from the group consisting of clouds, a water surface, and tree leaves.
35. A storage medium according to claim 30 , wherein
said spatial distribution state of said dot-sequences on said two-dimensional screen is represented by a probability density function on said two-dimensional screen, and
said spatial distribution state of said dot-sequences in said three-dimensional space is represented by a probability density function in said three-dimensional space.
36. A storage medium upon which a program is stored, said program being operable to cause a processing system to project a plurality of dot-sequences existing in a three-dimensional space upon a hypothetical two-dimensional screen by performing steps comprising:
determining a spatial distribution state of said dot-sequences on said two-dimensional screen using uniform random numbers, given that said plurality of dot-sequences randomly exist in a domain within said three-dimensional space; and
generating an image of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said dot-sequences on said two-dimensional screen.
37. A storage medium upon which a program is stored, said program being operable to cause a processing system to project a plurality of dot-sequences existing in a three-dimensional space upon a hypothetical two-dimensional screen by performing steps comprising:
determining a spatial distribution state of said dot-sequences on said two-dimensional screen based on a one-dimensional Gaussian distribution generated from uniform random numbers using the Box-Muller method, given that said plurality of dot-sequences randomly exist in a domain within said three-dimensional space; and
generating an image of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said dot-sequences on said two-dimensional screen.
38. A storage medium upon which a program is stored, said program being operable to cause a processing system to project a plurality of dot-sequences existing in a three-dimensional space upon a hypothetical two-dimensional screen by performing steps comprising:
determining a spatial distribution state of said dot-sequences on said two-dimensional screen using a two-dimensional Gaussian distribution, given that said plurality of dot-sequences randomly exist in a domain within said three-dimensional space; and
generating an image of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said dot-sequences on said two-dimensional screen.
39. A storage medium according to claim 38 , wherein said distribution state of said dot-sequences on said two-dimensional screen is a probability density function g(x, y) expressed as follows:
g ( x,y )=(1/2π)exp{−( x 2 +y 2 )/2},
wherein
said three-dimensional space is defined x, y, and z coordinate axes; and
said two-dimensional screen is oriented along the x and y axes.
40. An apparatus for projecting a plurality of dot-sequences existing in a three-dimensional space upon a hypothetical two-dimensional screen, said apparatus comprising:
a graphics data processor operable to (i) obtain a mathematical expression of a spatial distribution state of said dot-sequences in said three-dimensional space, given that said plurality of dot-sequences randomly exist in a domain within said three-dimensional space, and (ii) determine a mathematical expression of a spatial distribution state of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said plurality of dot-sequences in said three-dimensional space; and
a rendering processor operable to generate an image of said dot-sequences on said two-dimensional screen based on said mathematical expression of said spatial distribution state of said dot-sequences on said two-dimensional screen.
41. An apparatus according to claim 40 , wherein:
said graphics data processor is further operable to (i) determine a point of view within said three-dimensional space from which to provide a reference for projecting said dot-sequences onto said two-dimensional screen, (ii) determine a direction of view from said point of view toward said plurality of dot-sequences; and
said rendering processor is further operable to generate said image of said dot-sequences on said two-dimensional screen based on said mathematical expression of said spatial distribution state of said dot-sequences on said two-dimensional screen.
42. An apparatus according to claim 40 , wherein said mathematical expression of said spatial distribution state of said dot-sequences on said two-dimensional screen is determined using uniform random numbers.
43. An apparatus according to claim 40 , wherein said mathematical expression of said spatial distribution state of said dot-sequences on said two-dimensional screen is determined using a one-dimensional Gaussian distribution generated from uniform random numbers using the Box-Muller method.
44. An apparatus according to claim 40 , wherein said mathematical expression of said spatial distribution state of said dot-sequences on said two-dimensional screen is determined using a two-dimensional Gaussian distribution.
45. An apparatus according to claim 40 , wherein:
said mathematical expression of said spatial distribution state of said dot-sequences in said three-dimensional space is a probability density function in said three-dimensional space; and
said mathematical expression of said spatial distribution state of said dot-sequences on said two-dimensional screen is a probability density function on said two-dimensional screen.
46. A method for projecting a plurality of dot-sequences existing in a three-dimensional space upon a hypothetical two-dimensional screen, said method comprising:
obtaining a mathematical expression of a spatial distribution state of said dot-sequences in said three-dimensional space, given that said plurality of dot-sequences randomly exist in a domain within said three-dimensional space;
determining a mathematical expression of a spatial distribution state of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said plurality of dot-sequences in said three-dimensional space; and
generating an image of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said dot-sequences on said two-dimensional screen.
47. A method according to claim 46 , further comprising:
determining a point of view within said three-dimensional space from which to provide a reference for projecting said dot-sequences onto said two-dimensional screen;
determining a direction of view from said point of view toward said plurality of dot-sequences; and
generating said image of said dot-sequences on said two-dimensional screen based on said mathematical expression of said spatial distribution state of said dot-sequences on said two-dimensional screen.
48. A method according to claim 46 , wherein said mathematical expression of said spatial distribution state of said dot-sequences on said two-dimensional screen is determined using uniform random numbers.
49. A method according to claim 46 , wherein said mathematical expression of said spatial distribution state of said dot-sequences on said two-dimensional screen is determined using a one-dimensional Gaussian distribution generated from uniform random numbers using the Box-Muller method.
50. A method according to claim 46 , wherein said mathematical expression of said spatial distribution state of said dot-sequences on said two-dimensional screen is determined using a two-dimensional Gaussian distribution.
51. A method according to claim 46 , wherein:
said mathematical expression of said spatial distribution state of said dot-sequences in said three-dimensional space is a probability density function in said three-dimensional space; and
said mathematical expression of said spatial distribution state of said dot-sequences on said two-dimensional screen is a probability density function on said two-dimensional screen.
52. A storage medium containing a program operable to cause a processor to perform actions for projecting a plurality of dot-sequences existing in a three-dimensional space upon a hypothetical two-dimensional screen, said actions comprising:
obtaining a mathematical expression of a spatial distribution state of said dot-sequences in said three-dimensional space, given that said plurality of dot-sequences randomly exist in a domain within said three-dimensional space;
determining a mathematical expression of a spatial distribution state of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said plurality of dot-sequences in said three-dimensional space; and
generating an image of said dot-sequences on said two-dimensional screen based on said spatial distribution state of said dot-sequences on said two-dimensional screen.
53. A recording medium according to claim 52 , further comprising:
determining a point of view within said three-dimensional space from which to provide a reference for projecting said dot-sequences onto said two-dimensional screen;
determining a direction of view from said point of view toward said plurality of dot-sequences; and
generating said image of said dot-sequences on said two-dimensional screen based on said mathematical expression of said spatial distribution state of said dot-sequences on said two-dimensional screen.
54. A recording medium according to claim 52 , wherein said mathematical expression of said spatial distribution state of said dot-sequences on said two-dimensional screen is determined using uniform random numbers.
55. A recording medium according to claim 52 , wherein said mathematical expression of said spatial distribution state of said dot-sequences on said two-dimensional screen is determined using a one-dimensional Gaussian distribution generated from uniform random numbers using the Box-Muller method.
56. A recording medium according to claim 52 , wherein said mathematical expression of said spatial distribution state of said dot-sequences on said two-dimensional screen is determined using a two-dimensional Gaussian distribution.
57. A recording medium according to claim 52 , wherein:
said mathematical expression of said spatial distribution state of said dot-sequences in said three-dimensional space is a probability density function in said three-dimensional space; and
said mathematical expression of said spatial distribution state of said dot-sequences on said two-dimensional screen is a probability density function on said two-dimensional screen.
58. An apparatus, comprising:
a graphics processor operable to (i) receive a model of an object in three-dimensional space including a plurality of randomly distributed groups of one or more polygons, (ii) determine a first mathematical expression that represents said random distribution of said groups of one or more polygons in said three-dimensional space, and (iii) determine a second mathematical expression that represents a distribution of said groups of one or more polygons in a two-dimensional space based on said first mathematical expression; and
a rendering processor operable to render said groups of one or more polygons onto a two-dimensional plane of said two-dimensional space using said second mathematical expression.
59. An apparatus according to claim 58 , wherein:
said graphics data processor is further operable to (i) determine a point of view within said three-dimensional space from which to provide a reference for projecting said polygons onto said two-dimensional screen, (ii) determine a direction of view from said point of view toward said polygons; and
said rendering processor is further operable to generate said image of said polygons on said two-dimensional screen based on said second mathematical expression.
60. An apparatus according to claim 58 , wherein said second mathematical expression is determined using uniform random numbers.
61. An apparatus according to claim 58 , wherein said second mathematical expression is determined using a one-dimensional Gaussian distribution generated from uniform random numbers using the Box-Muller method.
62. An apparatus according to claim 58 , wherein said second mathematical expression is determined using a two-dimensional Gaussian distribution.
63. An apparatus according to claim 58 , wherein:
said first mathematical expression of said spatial distribution state of said polygons in said three-dimensional space is a probability density function in said three-dimensional space; and
said second mathematical expression of said spatial distribution state of said polygons on said two-dimensional screen is a probability density function on said two-dimensional screen.
64. A method, comprising:
providing a model of an object in three-dimensional space including a plurality of randomly distributed groups of one or more polygons;
determining a first mathematical expression that represents said random distribution of said groups of one or more polygons in said three-dimensional space;
determining a second mathematical expression that represents a distribution of said groups of one or more polygons in a two-dimensional space based on said first mathematical expression; and
rendering said groups of one or more polygons onto a two-dimensional plane of said two-dimensional space using said second mathematical expression.
65. A method according to claim 64 , further comprising:
determining a point of view within said three-dimensional space from which to provide a reference for projecting said polygons onto said two-dimensional screen;
determining a direction of view from said point of view toward said plurality of polygons; and
generating said image of said polygons on said two-dimensional screen based on said second mathematical expression.
66. A method according to claim 64 , wherein said second mathematical expression is determined using uniform random numbers.
67. A method according to claim 64 , wherein said second mathematical expression is determined using a one-dimensional Gaussian distribution generated from uniform random numbers using the Box-Muller method.
68. A method according to claim 64 , wherein said second mathematical expression is determined using a two-dimensional Gaussian distribution.
69. A method according to claim 64 , wherein:
said first mathematical expression of said spatial distribution state of said polygons in said three-dimensional space is a probability density function in said three-dimensional space; and
said second mathematical expression of said spatial distribution state of said polygons on said two-dimensional screen is a probability density function on said two-dimensional screen.
70. A storage medium containing a program operable to cause a processor to perform actions, said actions comprising:
receiving a model of an object in three-dimensional space including a plurality of randomly distributed groups of one or more polygons;
determining a first mathematical expression that represents said random distribution of said groups of one or more polygons in said three-dimensional space;
determining a second mathematical expression that represents a distribution of said groups of one or more polygons in a two-dimensional space based on said first mathematical expression; and
rendering said groups of one or more polygons onto a two-dimensional plane of said two-dimensional space using said second mathematical expression.
71. A storage medium according to claim 70 , further comprising:
determining a point of view within said three-dimensional space from which to provide a reference for projecting said polygons onto said two-dimensional screen;
determining a direction of view from said point of view toward said plurality of polygons; and
generating said image of said polygons on said two-dimensional screen based on said second mathematical expression.
72. A storage medium according to claim 70 , wherein said second mathematical expression is determined using uniform random numbers.
73. A storage medium according to claim 70 , wherein said second mathematical expression is determined using a one-dimensional Gaussian distribution generated from uniform random numbers using the Box-Muller method.
74. A storage medium according to claim 70 , wherein said second mathematical expression is determined using a two-dimensional Gaussian distribution.
75. A storage medium according to claim 70 , wherein:
said first mathematical expression of said spatial distribution state of said polygons in said three-dimensional space is a probability density function in said three-dimensional space; and
said second mathematical expression of said spatial distribution state of said polygons on said two-dimensional screen is a probability density function on said two-dimensional screen.