In this paper we present a simple and efficient algorithm for generating uniformaly distributed samples on the unit sphere based on an Archimedes\u27 theorem. The implementation is straightforward and may be easily extended to include stratified sampling for variance reduction. Applications in image synthesis include solid angle measurement, irradiance computation, and rendering equation solution for geometrically complex environments

Badler, Norman

Shao, Min-Zhi

English

Kosmopolis

University of Pennsylvania 
ScholarlyCommons 
Technical Reports (CIS) Department of Computer & Information Science 
1-1-1996 
Spherical Sampling by Archimedes' Theorem 
Min-Zhi Shao 
University of Pennsylvania 
Norman Badler 
University of Pennsylvania 
Follow this and additional works at: https://repository.upenn.edu/cis_reports 
 Part of the Theory and Algorithms Commons 
Recommended Citation 
Min-Zhi Shao and Norman Badler, "Spherical Sampling by Archimedes' Theorem", . January 1996. 
University of Pennsylvania Department of Computer and Information Science Technical Report No.MS-CIS-96-02. 
This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_reports/184 
For more information, please contact repository@pobox.upenn.edu. 
Spherical Sampling by Archimedes' Theorem 
Abstract 
In this paper we present a simple and efficient algorithm for generating uniformaly distributed samples on 
the unit sphere based on an Archimedes' theorem. The implementation is straightforward and may be 
easily extended to include stratified sampling for variance reduction. Applications in image synthesis 
include solid angle measurement, irradiance computation, and rendering equation solution for 
geometrically complex environments. 
Keywords 
Archiniedes' theorem, axial projection, Monte Carlo, solid angle, spherical sampling 
Disciplines 
Theory and Algorithms 
Comments 
University of Pennsylvania Department of Computer and Information Science Technical Report No.MS-
CIS-96-02. 
This technical report is available at ScholarlyCommons: https://repository.upenn.edu/cis_reports/184 
Spherical Sampling by Archimedes' Theorem 
MS-CIS-96-02 
Min-Zhi Shao 
Norman I. Badler 
University of Pennsylvania 
School of Engineering and Applied Science 
Compriter and Information Science Department 
Philadelphia, PA 19104-6380 
Spherical Sampling by Archimedes' Theorem 
Min-Zhi Shao and Norman I. Badler 
Department of Computer and Information Science 
University of Pennsylvania 
January 1996 
Technical Report MS-CIS-96-02 
Abstract 
In this paper we present a simple and efficient algorithm for generating uniformly 
distributed samples on the unit sphere based on an Archimedes' theorem. The imple- 
mentation is straightforward and may be easily extended to include stratified sampling 
for variance reduction. Applications in image synthesis include solid angle measurement, 
irradiance computation, and rendering equation solution for geometrically complex envi- 
ronments. 
CR Categories and Subject Descriptors: 1.3.5 [Computational Geometry and Object 
Modeling]: Geometric Algorithms; 1.3.7 [Computer Graphics]: Three-Dimensional Graphics 
and Realism. 
Additional Key Words and Phrases: Archiniedes' theorem, axial projection, Monte Carlo, 
solid angle, spherical sampling. 
1 Introduction 
In computer graphics, the Monte Carlo method [4] has been used in solving the rendering 
equations, evaluating the form factors, and elsewhere [5 ,  p. 1821. For Monte Carlo algo- 
rithms, random sampling is a fundamental operation. In image synthesis, an important 
sampling domain is the solid angles, i.e., regions on the unit sphere. Despite a significant 
amount of research in this area, to our knowledge, a uniformly distributed random sampling 
algorithm is not available for arbitrary measurable regions on the unit sphere. Recently, 
Arvo [2] presented such an algorithm for the spherical triangles. 
In this paper, we present a general spherical sampling algorithm based on an Archimedes' 
theorem on the sphere and cylinder. 
2 Archimedes' Theorem 
In his lifetime, the great Greek mathematician Archimedes (287-212 B.c.) had made many 
important discoveries. Among them are the calculation of n, the mechanics law of lever, and 
the hydrostatics law of floating body. The following is a theorem that in accordance with 
Archimedes' wishes was inscribed on his tombstone. It comes from a corollary in his work 
On the Sphere and Cylinder [3, p. 1071. 
Archimedes' Theorem (Global). The area of a sphere equals the area of every right 
circular cylinder circumscribed about the sphere excluding the bases. 
By the method of exhaustion, i.e., using inscribed and circumscribed rectilinear figures 
to "exhaust" the area, Archimedes proved that both areas are four times the area of the 
great circle on the sphere. We know that it is 4nr2 where r is the radius of the sphere. By 
more powerful techniques of calculus, however, this elegant global property turns out to be 
also local. That is, the same property holds for any point and its immediate neighborhood 
on the sphere. I t  is stated in the next theorem. 
Archimedes' Theorem (Local). The axial projection of any measurable region on a 
sphere on the right circular cylinder circumscribed about the sphere preserves area. 
Suppose P is a point on the sphere and M is the closest point to P on the axis of the 
circumscribed cylinder, the axial projection P' of P is the point a t  which ray M P  intersects 
the cylinder; see Figure 1. The axial projection S' of a region S on the sphere is the set of 
axial projections of all points of the region. The above theorem asserts that the areas of S 
and S' equal. 
It should be noted that the axial projection is a bijection for all points on the sphere 
except the two axial poles. For these two points, according to our definition, their axial 
projections are undetermined. We may, however, assume that they are projected to the 
entire corresponding base circles of the cylinder. Since the area of a circle is zero, the above 
theorem should hold if it holds anywhere else on the sphere. 
The proof of the theorem is elementary. We begin with a differential area dS on the 
sphere and its axial projection dS1 on the cylinder; See Figure 2. The differential area dS on 
the sphere is enclosed by differential arcs of two small circles with colatitudes 0 and B + d0 
and two great semicircles with longitudes 4 and Q + d$, where 0 < B < .~r and 0 < 4 < 2n. 
Easily, its axial projection dS' on the cylinder is enclosed by two differential circular arcs with 
heights r cos (0) and r cos (6 + $0) and two differential vertical line segments with longitudes 
q5 and 4 + I@, where r is the radius of the sphere. 
We have the differential area on the sphere 
dS = rd6 . r sin (6)clq + O(dOd4) 
= r2 sin (O)d0d+ + 0 (dB&$), 
and the differential area of its axial projection on the cylinder 
dS1 = r(cos (0) - cos (0 + dB)) - rd@ 
= r2(cos (8) - cos (0) cos (do) + sin ( H )  sin (cl8))d4 
= r2 sin (6)dOd4 + O(dOd$). 
For any measurable region S on the sphere and its axial projection $7' on the cylinder, 
their areas are Riemann-integrable [I, p. 3891. Hence, the theorem immediately follows by 
Equations (1) and (2). 
I t  is noted that the global theorem is a special case of the local theorem where the region 
,$ is the entire sphere and its axial projection S' is the entire cylinder excluding the bases. 
3 Spherical Sampling 
By the Archimedes' theorem, for any two measurable regions S1 and S2 on the unit sphere 
with equal areas, their axial projections Si and Si on the circumscribed cylinder will have 
equal areas. The reverse is also true because the axial projection is a bijection except for 
the two poles and their corresponding base circles which all have zero area. While the 
sphere is not developable, the cylinder is. This naturally leads to a spherical sampling 
algorithm which can be described in one sentence: Generate a random point on the cylinder 
[- 1,1] x [ 0 . 2 ~ ]  and then find its inverse axial projection on the unit sphere. If a random point 
is uniformly distributed on the cylinder, by the above argument, its inverse axial projection 
will be uniformly distributed on the sphere. 
Some final remarks may be in order. 
1. The algorithm may be used to sample or estimate the area of any measurable region 
on the unit sphere to any precision provided a point-in-region test algorithm exists. 
Similarly, it may be used to sample or estimate the solid angle subtended by any 
measurable arbitrarily-shaped object to any precision provided a line-object stabbing 
algorithm exists. For efficiency, bounding boxes may be constructed in the domain 
[- 1,1] x [O,2.ir] for subsets on the unit sphere. 
2. Stratified sampling is trivial over domain [-1,1] x [O, 2 ~ 1 .  I t  is recommended since 
stratification usually reduces variance and generates more evenly distributed samples. 
3. Numerical truncation errors are not uniformly distributed over the unit sphere under 
inverse axial projection. They tend to be greater around the pole areas. One solution 
is to use a random axis for every sample. For stratified sampling under this scheme, 
we may multiply a stratified axial sampling with a stratified cylindrical sampling over 
possibly coarser partition grids. 
Figures 3 and 4 show examples of uniform and stratified samplings on a sphere and 
a spherical triangle. Figure 5 shows samples over the solid angle subtended by a teapot. 
A pseudo-random double-precision floating-point number generator has been used in the 
implementation. 
4 Conclusions 
Based on an Archimedes' theorem, we have presented a simple and efficient algorithm 
for generating uniformly distributed samples on the unit sphere. The implementation is 
straightforward and may be easily extended to include stratified sampling for variance 
reduction. Applications in image synthesis include solid angle measurement, irradiance 
computation, and rendering equation solution for geometrically complex environments. 
References 
[I] T. M. Apostol, Mathematical Analysis, Addison-Wesley, Reading, Massachusetts, 1974. 
[2] J. Arvo, "Stratified sampling of spherical triangles," Proc. ACM Siggraph, 437438, 
August 1995. 
[31 M. Kline, Mathematical Thought from Ancient to Modern Emes, Oxford University 
Press, New York, 1972. 
[4] R. Y. Rubinstein, Simulation and the Monte Carlo Method, Wiley, New York, 1981. 
[5] F. X. Sillion and C. Puech, Radiosity and Global Illumination, Morgan Kaufmann, San 
Francisco, 1994. 
Figure 1: A sphere and a right circular cylinder circumscribed about the sphere. Spherical 
point P and region S and their axial projections Pf  and S f  on the cylinder. 
Figure 2: Differential area dS on the sphere and its axial projection dSf  on the cylinder. 
(a) uniform sampling 
(b) stratified sampling 
Figure 3: Samples on a sphere. 
6 
(a) uniform sampling 
(b) stratified sampling 
Figure 4: Samples on a spherical triangle. 
(a) uniform sampling 
(b) stratified sampling 
Figure 5: Samples over the solid angle subtended by a teapot. 
8 


Spherical Sampling by Archimedes\u27 Theorem

ScholarlyCommons@Penn

University of Pennsylvania ScholarlyCommons Technical Reports (CIS) Department of Computer & Information Science 1-1-1996 Spherical Sampling by Archimedes' Theorem Min-Zhi Shao University of Pennsylvania Norman Badler University of Pennsylvania Follow this and additional works at: https://repository.upenn.edu/cis_reports  Part of the Theory and Algorithms Commons Recommended Citation Min-Zhi Shao and Norman Badler, "Spherical Sampling by Archimedes' Theorem", . January 1996. University of Pennsylvania Department of Computer and Information Science Technical Report No.MS-CIS-96-02. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_reports/184 For more information, please contact repository@pobox.upenn.edu. Spherical Sampling by Archimedes' Theorem Abstract In this paper we present a simple and efficient algorithm for generating uniformaly distributed samples on the unit sphere based on an Archimedes' theorem. The implementation is straightforward and may be easily extended to include stratified sampling for variance reduction. Applications in image synthesis include solid angle measurement, irradiance computation, and rendering equation solution for geometrically complex environments. Keywords Archiniedes' theorem, axial projection, Monte Carlo, solid angle, spherical sampling Disciplines Theory and Algorithms Comments University of Pennsylvania Department of Computer and Information Science Technical Report No.MS-CIS-96-02. This technical report is available at ScholarlyCommons: https://repository.upenn.edu/cis_reports/184 Spherical Sampling by Archimedes' Theorem MS-CIS-96-02 Min-Zhi Shao Norman I. Badler University of Pennsylvania School of Engineering and Applied Science Compriter and Information Science Department Philadelphia, PA 19104-6380 Spherical Sampling by Archimedes' Theorem Min-Zhi Shao and Norman I. Badler Department of Computer and Information Science University of Pennsylvania January 1996 Technical Report MS-CIS-96-02 Abstract In this paper we present a simple and efficient algorithm for generating uniformly distributed samples on the unit sphere based on an Archimedes' theorem. The imple- mentation is straightforward and may be easily extended to include stratified sampling for variance reduction. Applications in image synthesis include solid angle measurement, irradiance computation, and rendering equation solution for geometrically complex envi- ronments. CR Categories and Subject Descriptors: 1.3.5 [Computational Geometry and Object Modeling]: Geometric Algorithms; 1.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism. Additional Key Words and Phrases: Archiniedes' theorem, axial projection, Monte Carlo, solid angle, spherical sampling. 1 Introduction In computer graphics, the Monte Carlo method [4] has been used in solving the rendering equations, evaluating the form factors, and elsewhere [5 ,  p. 1821. For Monte Carlo algo- rithms, random sampling is a fundamental operation. In image synthesis, an important sampling domain is the solid angles, i.e., regions on the unit sphere. Despite a significant amount of research in this area, to our knowledge, a uniformly distributed random sampling algorithm is not available for arbitrary measurable regions on the unit sphere. Recently, Arvo [2] presented such an algorithm for the spherical triangles. In this paper, we present a general spherical sampling algorithm based on an Archimedes' theorem on the sphere and cylinder. 2 Archimedes' Theorem In his lifetime, the great Greek mathematician Archimedes (287-212 B.c.) had made many important discoveries. Among them are the calculation of n, the mechanics law of lever, and the hydrostatics law of floating body. The following is a theorem that in accordance with Archimedes' wishes was inscribed on his tombstone. It comes from a corollary in his work On the Sphere and Cylinder [3, p. 1071. Archimedes' Theorem (Global). The area of a sphere equals the area of every right circular cylinder circumscribed about the sphere excluding the bases. By the method of exhaustion, i.e., using inscribed and circumscribed rectilinear figures to "exhaust" the area, Archimedes proved that both areas are four times the area of the great circle on the sphere. We know that it is 4nr2 where r is the radius of the sphere. By more powerful techniques of calculus, however, this elegant global property turns out to be also local. That is, the same property holds for any point and its immediate neighborhood on the sphere. I t  is stated in the next theorem. Archimedes' Theorem (Local). The axial projection of any measurable region on a sphere on the right circular cylinder circumscribed about the sphere preserves area. Suppose P is a point on the sphere and M is the closest point to P on the axis of the circumscribed cylinder, the axial projection P' of P is the point a t  which ray M P  intersects the cylinder; see Figure 1. The axial projection S' of a region S on the sphere is the set of axial projections of all points of the region. The above theorem asserts that the areas of S and S' equal. It should be noted that the axial projection is a bijection for all points on the sphere except the two axial poles. For these two points, according to our definition, their axial projections are undetermined. We may, however, assume that they are projected to the entire corresponding base circles of the cylinder. Since the area of a circle is zero, the above theorem should hold if it holds anywhere else on the sphere. The proof of the theorem is elementary. We begin with a differential area dS on the sphere and its axial projection dS1 on the cylinder; See Figure 2. The differential area dS on the sphere is enclosed by differential arcs of two small circles with colatitudes 0 and B + d0 and two great semicircles with longitudes 4 and Q + d$, where 0 < B < .~r and 0 < 4 < 2n. Easily, its axial projection dS' on the cylinder is enclosed by two differential circular arcs with heights r cos (0) and r cos (6 + $0) and two differential vertical line segments with longitudes q5 and 4 + I@, where r is the radius of the sphere. We have the differential area on the sphere dS = rd6 . r sin (6)clq + O(dOd4) = r2 sin (O)d0d+ + 0 (dB&$), and the differential area of its axial projection on the cylinder dS1 = r(cos (0) - cos (0 + dB)) - rd@ = r2(cos (8) - cos (0) cos (do) + sin ( H )  sin (cl8))d4 = r2 sin (6)dOd4 + O(dOd$). For any measurable region S on the sphere and its axial projection $7' on the cylinder, their areas are Riemann-integrable [I, p. 3891. Hence, the theorem immediately follows by Equations (1) and (2). I t  is noted that the global theorem is a special case of the local theorem where the region ,$ is the entire sphere and its axial projection S' is the entire cylinder excluding the bases. 3 Spherical Sampling By the Archimedes' theorem, for any two measurable regions S1 and S2 on the unit sphere with equal areas, their axial projections Si and Si on the circumscribed cylinder will have equal areas. The reverse is also true because the axial projection is a bijection except for the two poles and their corresponding base circles which all have zero area. While the sphere is not developable, the cylinder is. This naturally leads to a spherical sampling algorithm which can be described in one sentence: Generate a random point on the cylinder [- 1,1] x [ 0 . 2 ~ ]  and then find its inverse axial projection on the unit sphere. If a random point is uniformly distributed on the cylinder, by the above argument, its inverse axial projection will be uniformly distributed on the sphere. Some final remarks may be in order. 1. The algorithm may be used to sample or estimate the area of any measurable region on the unit sphere to any precision provided a point-in-region test algorithm exists. Similarly, it may be used to sample or estimate the solid angle subtended by any measurable arbitrarily-shaped object to any precision provided a line-object stabbing algorithm exists. For efficiency, bounding boxes may be constructed in the domain [- 1,1] x [O,2.ir] for subsets on the unit sphere. 2. Stratified sampling is trivial over domain [-1,1] x [O, 2 ~ 1 .  I t  is recommended since stratification usually reduces variance and generates more evenly distributed samples. 3. Numerical truncation errors are not uniformly distributed over the unit sphere under inverse axial projection. They tend to be greater around the pole areas. One solution is to use a random axis for every sample. For stratified sampling under this scheme, we may multiply a stratified axial sampling with a stratified cylindrical sampling over possibly coarser partition grids. Figures 3 and 4 show examples of uniform and stratified samplings on a sphere and a spherical triangle. Figure 5 shows samples over the solid angle subtended by a teapot. A pseudo-random double-precision floating-point number generator has been used in the implementation. 4 Conclusions Based on an Archimedes' theorem, we have presented a simple and efficient algorithm for generating uniformly distributed samples on the unit sphere. The implementation is straightforward and may be easily extended to include stratified sampling for variance reduction. Applications in image synthesis include solid angle measurement, irradiance computation, and rendering equation solution for geometrically complex environments. References [I] T. M. Apostol, Mathematical Analysis, Addison-Wesley, Reading, Massachusetts, 1974. [2] J. Arvo, "Stratified sampling of spherical triangles," Proc. ACM Siggraph, 437438, August 1995. [31 M. Kline, Mathematical Thought from Ancient to Modern Emes, Oxford University Press, New York, 1972. [4] R. Y. Rubinstein, Simulation and the Monte Carlo Method, Wiley, New York, 1981. [5] F. X. Sillion and C. Puech, Radiosity and Global Illumination, Morgan Kaufmann, San Francisco, 1994. Figure 1: A sphere and a right circular cylinder circumscribed about the sphere. Spherical point P and region S and their axial projections Pf  and S f  on the cylinder. Figure 2: Differential area dS on the sphere and its axial projection dSf  on the cylinder. (a) uniform sampling (b) stratified sampling Figure 3: Samples on a sphere. 6 (a) uniform sampling (b) stratified sampling Figure 4: Samples on a spherical triangle. (a) uniform sampling (b) stratified sampling Figure 5: Samples over the solid angle subtended by a teapot. 8 

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Spherical Sampling by Archimedes\u27 Theorem

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