-
Notifications
You must be signed in to change notification settings - Fork 41
/
sphere_quad.html
385 lines (346 loc) · 11.8 KB
/
sphere_quad.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
<html>
<head>
<title>
SPHERE_QUAD - Quadrature on the Surface of the Unit Sphere
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SPHERE_QUAD <br> Quadrature on the Surface of the Unit Sphere
</h1>
<hr>
<p>
<b>SPHERE_QUAD</b>
is a C++ library which
estimates the integral of a scalar function F(X,Y,Z) over the surface of
the unit sphere centered at the origin.
</p>
<p>
The library includes one function, <b>SPHERE01_QUAD_MC</b>, which
estimates the integral using a Monte Carlo approach. It randomly
samples points on the surface, and estimates the integral as the
average of these values times the area of the surface.
</p>
<p>
The library includes three functions based on the idea of a
latitudinal/longitudinal grid: <b>SPHERE01_QUAD_LLC</b>,
<b>SPHERE01_QUAD_LLV</b> and <b>SPHERE01_QUAD_LLM</b>.
The surface of the sphere is divided into rectangles whose sides
are always lines of latitude or longitude. Each rectangle is then
split diagonally into a pair of triangles (except for the degenerate
rectangles that include the north or south pole as a vertex.)
</p>
<p>
The user controls the accuracy of the integral estimate by specifying
a maximum side length H. The functions determine angular increments
that guarantee the size restriction. Of course, this means that the
restriction on latitude, enforced at the equator, will result in excessively
small triangles away from the equator. That is a penalty of
using this simple subdivision scheme.
</p>
<p>
The library includes three functions based on the idea of first
subdividing the surface into 20 congruent spherical triangles, based
on the projection of a regular icosahedron. The functions
<b>SPHERE01_QUAD_ICOS1C</b>, <b>SPHERE01_QUAD_ICOS1V</b> and
<b>SPHERE01_QUAD_ICOS1M</b> use this idea, along with subdivision.
</p>
<p>
The function <b>SPHERE01_QUAD_ICOS2V</b> is similar to
<b>SPHERE01_QUAD_ICOS1V</b> but uses a more sophisticated algorithm
to project points from the planar triangle to the unit sphere.
However, the modifications seem to make little difference to the
resulting integral estimate.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>SPHERE_QUAD</b> is available in
<a href = "../../c_src/sphere_quad/sphere_quad.html">a C version</a> and
<a href = "../../cpp_src/sphere_quad/sphere_quad.html">a C++ version</a> and
<a href = "../../f77_src/sphere_quad/sphere_quad.html">a FORTRAN77 version</a> and
<a href = "../../f_src/sphere_quad/sphere_quad.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sphere_quad/sphere_quad.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/random_data/random_data.html">
RANDOM_DATA</a>,
a C++ library which
generates sample points for
various probability distributions, spatial dimensions, and geometries;
</p>
<p>
<a href = "../../cpp_src/sphere_exactness/sphere_exactness.html">
SPHERE_EXACTNESS</a>,
a C++ program which
tests the polynomial exactness of a quadrature rule for the unit sphere;
</p>
<p>
<a href = "../../cpp_src/sphere_grid/sphere_grid.html">
SPHERE_GRID</a>,
a C++ library which
provides a number of ways of generating grids of points, or of
points and lines, or of points and lines and faces, over the unit sphere.
</p>
<p>
<a href = "../../cpp_src/sphere_lebedev_rule/sphere_lebedev_rule.html">
SPHERE_LEBEDEV_RULE</a>,
a C++ library which
computes Lebedev quadrature rules for the unit sphere;
</p>
<p>
<a href = "../../cpp_src/sphere_monte_carlo/sphere_monte_carlo.html">
SPHERE_MONTE_CARLO</a>,
a C++ library which
applies a Monte Carlo method to estimate the integral of a function
over the surface of the unit sphere in 3D;
</p>
<p>
<a href = "../../cpp_src/sphere_triangle_quad/sphere_triangle_quad.html">
SPHERE_TRIANGLE_QUAD</a>,
a C++ library which
estimates the integral of a function over a spherical triangle.
</p>
<p>
<a href = "../../cpp_src/stroud/stroud.html">
STROUD</a>,
a C++ library which
approximates the integral of a function on the surface or in the interior
of a variety of geometric shapes.
</p>
<p>
<a href = "../../m_src/xyz_display/xyz_display.html">
XYZ_DISPLAY</a>,
a MATLAB program which
reads XYZ information defining points in 3D,
and displays an image in the MATLAB graphics window.
</p>
<p>
<a href = "../../cpp_src/xyz_display_opengl/xyz_display_opengl.html">
XYZ_DISPLAY_OPENGL</a>,
a C++ program which
reads XYZ information defining points in 3D,
and displays an image using OpenGL.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ul>
<li>
James Arvo,<br>
Stratified sampling of spherical triangles,<br>
Computer Graphics Proceedings, Annual Conference Series, <br>
ACM SIGGRAPH '95, pages 437-438, 1995.
</li>
<li>
Philip Davis, Philip Rabinowitz,<br>
Methods of Numerical Integration,<br>
Second Edition,<br>
Dover, 2007,<br>
ISBN: 0486453391,<br>
LC: QA299.3.D28.
</li>
<li>
Jacob Goodman, Joseph ORourke, editors,<br>
Handbook of Discrete and Computational Geometry,<br>
Second Edition,<br>
CRC/Chapman and Hall, 2004,<br>
ISBN: 1-58488-301-4,<br>
LC: QA167.H36.
</li>
</ul>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "sphere_quad.cpp">sphere_quad.cpp</a>, the source code.
</li>
<li>
<a href = "sphere_quad.hpp">sphere_quad.hpp</a>, the include file.
</li>
<li>
<a href = "sphere_quad.sh">sphere_quad.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "sphere_quad_prb.cpp">sphere_quad_prb.cpp</a>,
a sample problem.
</li>
<li>
<a href = "sphere_quad_prb.sh">sphere_quad_prb.sh</a>,
commands to compile and run the problem.
</li>
<li>
<a href = "sphere_quad_prb_output.txt">sphere_quad_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>ARC_COSINE</b> computes the arc cosine function, with argument truncation.
</li>
<li>
<b>ARC_SINE</b> computes the arc sine function, with argument truncation.
</li>
<li>
<b>ATAN4</b> computes the inverse tangent of the ratio Y / X.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the smaller of two I4's.
</li>
<li>
<b>I4VEC_COPY</b> copies an I4VEC.
</li>
<li>
<b>ICOS_SHAPE</b> describes a icosahedron.
</li>
<li>
<b>ICOS_SIZE</b> gives "sizes" for an icosahedron in 3D.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_GAMMA</b> evaluates Gamma(X) for a real argument.
</li>
<li>
<b>R8_MAX</b> returns the maximum of two R8's.
</li>
<li>
<b>R8_MIN</b> returns the minimum of two R8's.
</li>
<li>
<b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
</li>
<li>
<b>R8VEC_COPY</b> copies an R8VEC.
</li>
<li>
<b>R8VEC_DOT_PRODUCT</b> computes the dot product of a pair of R8VEC's.
</li>
<li>
<b>R8VEC_NORM</b> returns the L2 norm of an R8VEC.
</li>
<li>
<b>R8VEC_POLARIZE</b> decomposes an R8VEC into normal and parallel components.
</li>
<li>
<b>R8VEC_SUM</b> returns the sum of an R8VEC.
</li>
<li>
<b>SPHERE01_DISTANCE_XYZ</b> computes great circle distances on a unit sphere.
</li>
<li>
<b>SPHERE01_MONOMIAL_INTEGRAL</b> returns monomial integrals on the unit sphere.
</li>
<li>
<b>SPHERE01_QUAD_ICOS1C:</b> centroid rule, subdivide then project.
</li>
<li>
<b>SPHERE01_QUAD_ICOS1M:</b> midside rule, subdivide then project.
</li>
<li>
<b>SPHERE01_QUAD_ICOS1V:</b> vertex rule, subdivide then project.
</li>
<li>
<b>SPHERE01_QUAD_ICOS2V:</b> vertex rule, subdivide then project.
</li>
<li>
<b>SPHERE01_QUAD_LLC:</b> Longitude/Latitude grid with centroid rule.
</li>
<li>
<b>SPHERE01_QUAD_LLM:</b> longitude/latitude grid plus midside rule.
</li>
<li>
<b>SPHERE01_QUAD_LLV:</b> longitude/latitude grid with vertex rule.
</li>
<li>
<b>SPHERE01_QUAD_MC</b> uses the Monte Carlo rule for sphere quadrature.
</li>
<li>
<b>SPHERE01_QUAD_MC_SIZE</b> sizes a Monte Carlo rule for sphere quadrature.
</li>
<li>
<b>SPHERE01_SAMPLE</b> picks random points on a unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_ANGLES_TO_AREA:</b> area of a spherical triangle on the unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_PROJECT</b> projects from a plane triangle to a spherical triangle.
</li>
<li>
<b>SPHERE01_TRIANGLE_PROJECT2</b> projects from a plane triangle to a spherical triangle.
</li>
<li>
<b>SPHERE01_TRIANGLE_SAMPLE:</b> sample points from triangle on unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_SIDES_TO_ANGLES:</b> angles of spherical triangle on unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_VERTICES_TO_ANGLES:</b> angles of spherical triangle on unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_VERTICES_TO_AREA:</b> area of a spherical triangle on unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_VERTICES_TO_CENTROID:</b> centroid of spherical triangle on unit sphere.
</li>
<li>
<b>SPHERE01_TRIANGLE_VERTICES_TO_MIDPOINTS</b> gets the midsides of a spherical triangle.
</li>
<li>
<b>SPHERE01_TRIANGLE_VERTICES_TO_SIDES_3D:</b> sides of spherical triangle on unit sphere.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TP_TO_XYZ</b> converts unit spherical TP coordinates to XYZ coordinates.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 23 September 2010.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial html skeleton created by HTMLINDEX. -->
</html>