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annotate doc/interpreter/geometry.txi @ 8920:eb63fbe60fab
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| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Sat, 07 Mar 2009 10:41:27 -0500 |
| parents | 8ae26422a6ce |
| children | e9dc2ed2ec0f |
| rev | line source |
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| 8920 | 1 @c Copyright (C) 2007, 2008, 2009 John W. Eaton and David Bateman |
| 7018 | 2 @c |
| 3 @c This file is part of Octave. | |
| 4 @c | |
| 5 @c Octave is free software; you can redistribute it and/or modify it | |
| 6 @c under the terms of the GNU General Public License as published by the | |
| 7 @c Free Software Foundation; either version 3 of the License, or (at | |
| 8 @c your option) any later version. | |
| 9 @c | |
| 10 @c Octave is distributed in the hope that it will be useful, but WITHOUT | |
| 11 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
| 12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License | |
| 13 @c for more details. | |
| 14 @c | |
| 15 @c You should have received a copy of the GNU General Public License | |
| 16 @c along with Octave; see the file COPYING. If not, see | |
| 17 @c <http://www.gnu.org/licenses/>. | |
| 6558 | 18 |
| 19 @node Geometry | |
| 20 @chapter Geometry | |
| 21 | |
|
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22 Much of geometry code in Octave is based on the QHull library@footnote{Barber, |
| 6832 | 23 C.B., Dobkin, D.P., and Huhdanpaa, H.T., "The Quickhull algorithm for |
| 24 convex hulls," ACM Trans. on Mathematical Software, 22(4):469-483, Dec | |
| 25 1996, @url{http://www.qhull.org}}. Some of the documentation for Qhull, | |
| 26 particularly for the options that can be passed to @code{delaunay}, | |
| 27 @code{voronoi} and @code{convhull}, etc, is relevant to Octave users. | |
| 28 | |
| 6823 | 29 @menu |
| 30 * Delaunay Triangulation:: | |
| 31 * Voronoi Diagrams:: | |
| 32 * Convex Hull:: | |
| 33 * Interpolation on Scattered Data:: | |
| 34 @end menu | |
| 35 | |
| 36 @node Delaunay Triangulation | |
| 37 @section Delaunay Triangulation | |
| 38 | |
| 6832 | 39 The Delaunay triangulation is constructed from a set of |
| 40 circum-circles. These circum-circles are chosen so that there are at | |
| 41 least three of the points in the set to triangulation on the | |
| 42 circumference of the circum-circle. None of the points in the set of | |
| 43 points falls within any of the circum-circles. | |
| 44 | |
| 45 In general there are only three points on the circumference of any | |
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46 circum-circle. However, in some cases, and in particular for the |
| 6832 | 47 case of a regular grid, 4 or more points can be on a single |
| 48 circum-circle. In this case the Delaunay triangulation is not unique. | |
| 49 | |
| 6823 | 50 @DOCSTRING(delaunay) |
| 51 | |
| 52 The 3- and N-dimensional extension of the Delaunay triangulation are | |
| 53 given by @code{delaunay3} and @code{delaunayn} respectively. | |
| 54 @code{delaunay3} returns a set of tetrahedra that satisfy the | |
| 55 Delaunay circum-circle criteria. Similarly, @code{delaunayn} returns the | |
| 56 N-dimensional simplex satisfying the Delaunay circum-circle criteria. | |
| 7007 | 57 The N-dimensional extension of a triangulation is called a tessellation. |
| 6823 | 58 |
| 59 @DOCSTRING(delaunay3) | |
| 60 | |
| 61 @DOCSTRING(delaunayn) | |
| 62 | |
| 6832 | 63 An example of a Delaunay triangulation of a set of points is |
| 64 | |
| 65 @example | |
| 66 @group | |
| 67 rand ("state", 2); | |
| 68 x = rand (10, 1); | |
| 69 y = rand (10, 1); | |
| 70 T = delaunay (x, y); | |
| 71 X = [ x(T(:,1)); x(T(:,2)); x(T(:,3)); x(T(:,1)) ]; | |
| 72 Y = [ y(T(:,1)); y(T(:,2)); y(T(:,3)); y(T(:,1)) ]; | |
| 73 axis ([0, 1, 0, 1]); | |
| 74 plot(X, Y, "b", x, y, "r*"); | |
| 75 @end group | |
| 76 @end example | |
| 77 | |
| 6855 | 78 @ifset HAVE_QHULL |
| 6832 | 79 @ifnotinfo |
| 80 @noindent | |
| 81 The result of which can be seen in @ref{fig:delaunay}. | |
| 82 | |
| 83 @float Figure,fig:delaunay | |
| 84 @image{delaunay,8cm} | |
| 85 @caption{Delaunay triangulation of a random set of points} | |
| 86 @end float | |
| 87 @end ifnotinfo | |
| 6855 | 88 @end ifset |
| 6832 | 89 |
| 6823 | 90 @menu |
| 6832 | 91 * Plotting the Triangulation:: |
| 6823 | 92 * Identifying points in Triangulation:: |
| 93 @end menu | |
| 94 | |
| 6832 | 95 @node Plotting the Triangulation |
| 96 @subsection Plotting the Triangulation | |
| 97 | |
| 98 Octave has the functions @code{triplot} and @code{trimesh} to plot the | |
| 99 Delaunay triangulation of a 2-dimensional set of points. | |
| 100 | |
| 101 @DOCSTRING(triplot) | |
| 102 | |
| 103 @DOCSTRING(trimesh) | |
| 104 | |
| 105 The difference between @code{triplot} and @code{trimesh} is that the | |
| 106 former only plots the 2-dimensional triangulation itself, whereas the | |
| 107 second plots the value of some function @code{f (@var{x}, @var{y})}. | |
| 108 An example of the use of the @code{triplot} function is | |
| 109 | |
| 110 @example | |
| 111 @group | |
| 112 rand ("state", 2) | |
| 113 x = rand (20, 1); | |
| 114 y = rand (20, 1); | |
| 115 tri = delaunay (x, y); | |
| 116 triplot (tri, x, y); | |
| 117 @end group | |
| 118 @end example | |
| 119 | |
| 120 that plot the Delaunay triangulation of a set of random points in | |
| 121 2-dimensions. | |
| 122 @ifnotinfo | |
| 123 The output of the above can be seen in @ref{fig:triplot}. | |
| 124 | |
| 125 @float Figure,fig:triplot | |
| 126 @image{triplot,8cm} | |
| 127 @caption{Delaunay triangulation of a random set of points} | |
| 128 @end float | |
| 129 @end ifnotinfo | |
| 130 | |
| 6823 | 131 @node Identifying points in Triangulation |
| 132 @subsection Identifying points in Triangulation | |
| 133 | |
| 134 It is often necessary to identify whether a particular point in the | |
| 7007 | 135 N-dimensional space is within the Delaunay tessellation of a set of |
| 8480 | 136 points in this N-dimensional space, and if so which N-simplex contains |
| 7007 | 137 the point and which point in the tessellation is closest to the desired |
| 6823 | 138 point. The functions @code{tsearch} and @code{dsearch} perform this |
| 139 function in a triangulation, and @code{tsearchn} and @code{dsearchn} in | |
| 7007 | 140 an N-dimensional tessellation. |
| 6823 | 141 |
| 142 To identify whether a particular point represented by a vector @var{p} | |
| 8480 | 143 falls within one of the simplices of an N-simplex, we can write the |
| 6823 | 144 Cartesian coordinates of the point in a parametric form with respect to |
| 8480 | 145 the N-simplex. This parametric form is called the Barycentric |
| 146 Coordinates of the point. If the points defining the N-simplex are given | |
| 6823 | 147 by @code{@var{N} + 1} vectors @var{t}(@var{i},:), then the Barycentric |
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148 coordinates defining the point @var{p} are given by |
| 6823 | 149 |
| 150 @example | |
| 151 @var{p} = sum (@var{beta}(1:@var{N}+1) * @var{t}(1:@var{N}+1),:) | |
| 152 @end example | |
| 153 | |
| 154 @noindent | |
| 155 where there are @code{@var{N} + 1} values @code{@var{beta}(@var{i})} | |
| 156 that together as a vector represent the Barycentric coordinates of the | |
| 157 point @var{p}. To ensure a unique solution for the values of | |
| 158 @code{@var{beta}(@var{i})} an additional criteria of | |
| 159 | |
| 160 @example | |
| 161 sum (@var{beta}(1:@var{N}+1)) == 1 | |
| 162 @end example | |
| 163 | |
| 164 @noindent | |
| 165 is imposed, and we can therefore write the above as | |
| 166 | |
| 167 @example | |
| 168 @var{p} - @var{t}(end, :) = @var{beta}(1:end-1) * (@var{t}(1:end-1, :) | |
| 169 - ones(@var{N}, 1) * @var{t}(end, :) | |
| 170 @end example | |
| 171 | |
| 172 @noindent | |
| 173 Solving for @var{beta} we can then write | |
| 174 | |
| 175 @example | |
| 176 @var{beta}(1:end-1) = (@var{p} - @var{t}(end, :)) / (@var{t}(1:end-1, :) | |
| 177 - ones(@var{N}, 1) * @var{t}(end, :)) | |
| 178 @var{beta}(end) = sum(@var{beta}(1:end-1)) | |
| 179 @end example | |
| 180 | |
| 181 @noindent | |
| 182 which gives the formula for the conversion of the Cartesian coordinates | |
| 183 of the point @var{p} to the Barycentric coordinates @var{beta}. An | |
| 184 important property of the Barycentric coordinates is that for all points | |
| 8480 | 185 in the N-simplex |
| 6823 | 186 |
| 187 @example | |
| 188 0 <= @var{beta}(@var{i}) <= 1 | |
| 189 @end example | |
| 190 | |
| 191 @noindent | |
| 192 Therefore, the test in @code{tsearch} and @code{tsearchn} essentially | |
| 193 only needs to express each point in terms of the Barycentric coordinates | |
| 8480 | 194 of each of the simplices of the N-simplex and test the values of |
| 6823 | 195 @var{beta}. This is exactly the implementation used in |
| 196 @code{tsearchn}. @code{tsearch} is optimized for 2-dimensions and the | |
| 197 Barycentric coordinates are not explicitly formed. | |
| 198 | |
| 199 @DOCSTRING(tsearch) | |
| 200 | |
| 201 @DOCSTRING(tsearchn) | |
| 202 | |
| 203 An example of the use of @code{tsearch} can be seen with the simple | |
| 204 triangulation | |
| 205 | |
| 206 @example | |
| 207 @group | |
| 208 @var{x} = [-1; -1; 1; 1]; | |
| 209 @var{y} = [-1; 1; -1; 1]; | |
| 210 @var{tri} = [1, 2, 3; 2, 3, 1]; | |
| 211 @end group | |
| 212 @end example | |
| 213 | |
| 214 @noindent | |
| 215 consisting of two triangles defined by @var{tri}. We can then identify | |
| 216 which triangle a point falls in like | |
| 217 | |
| 218 @example | |
| 219 @group | |
| 220 tsearch (@var{x}, @var{y}, @var{tri}, -0.5, -0.5) | |
| 221 @result{} 1 | |
| 222 tsearch (@var{x}, @var{y}, @var{tri}, 0.5, 0.5) | |
| 223 @result{} 2 | |
| 224 @end group | |
| 225 @end example | |
| 226 | |
| 227 @noindent | |
| 228 and we can confirm that a point doesn't lie within one of the triangles like | |
| 229 | |
| 230 @example | |
| 231 @group | |
| 232 tsearch (@var{x}, @var{y}, @var{tri}, 2, 2) | |
| 233 @result{} NaN | |
| 234 @end group | |
| 235 @end example | |
| 236 | |
| 237 The @code{dsearch} and @code{dsearchn} find the closest point in a | |
| 238 tessellation to the desired point. The desired point does not | |
| 239 necessarily have to be in the tessellation, and even if it the returned | |
| 6832 | 240 point of the tessellation does not have to be one of the vertexes of the |
| 6823 | 241 N-simplex within which the desired point is found. |
| 242 | |
| 243 @DOCSTRING(dsearch) | |
| 244 | |
| 245 @DOCSTRING(dsearchn) | |
| 246 | |
| 247 An example of the use of @code{dsearch}, using the above values of | |
| 248 @var{x}, @var{y} and @var{tri} is | |
| 249 | |
| 250 @example | |
| 251 @group | |
| 252 dsearch (@var{x}, @var{y}, @var{tri}, -2, -2) | |
| 253 @result{} 1 | |
| 254 @end group | |
| 255 @end example | |
| 256 | |
| 257 If you wish the points that are outside the tessellation to be flagged, | |
| 258 then @code{dsearchn} can be used as | |
| 259 | |
| 260 @example | |
| 261 @group | |
| 262 dsearchn ([@var{x}, @var{y}], @var{tri}, [-2, -2], NaN) | |
| 263 @result{} NaN | |
| 264 dsearchn ([@var{x}, @var{y}], @var{tri}, [-0.5, -0.5], NaN) | |
| 265 @result{} 1 | |
| 266 @end group | |
| 267 @end example | |
| 268 | |
| 269 @noindent | |
| 270 where the point outside the tessellation are then flagged with @code{NaN}. | |
| 271 | |
| 272 @node Voronoi Diagrams | |
| 273 @section Voronoi Diagrams | |
| 274 | |
| 275 A Voronoi diagram or Voronoi tessellation of a set of points @var{s} in | |
| 276 an N-dimensional space, is the tessellation of the N-dimensional space | |
| 277 such that all points in @code{@var{v}(@var{p})}, a partitions of the | |
| 278 tessellation where @var{p} is a member of @var{s}, are closer to @var{p} | |
| 279 than any other point in @var{s}. The Voronoi diagram is related to the | |
| 6832 | 280 Delaunay triangulation of a set of points, in that the vertexes of the |
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281 Voronoi tessellation are the centers of the circum-circles of the |
| 6832 | 282 simplicies of the Delaunay tessellation. |
| 6823 | 283 |
| 284 @DOCSTRING(voronoi) | |
| 285 | |
| 286 @DOCSTRING(voronoin) | |
| 287 | |
| 6832 | 288 An example of the use of @code{voronoi} is |
| 289 | |
| 290 @example | |
| 291 @group | |
| 292 rand("state",9); | |
| 293 x = rand(10,1); | |
| 294 y = rand(10,1); | |
| 295 tri = delaunay (x, y); | |
| 296 [vx, vy] = voronoi (x, y, tri); | |
| 297 triplot (tri, x, y, "b"); | |
| 298 hold on; | |
| 299 plot (vx, vy, "r"); | |
| 300 @end group | |
| 301 @end example | |
| 302 | |
| 6855 | 303 @ifset HAVE_QHULL |
| 6832 | 304 @ifnotinfo |
| 305 @noindent | |
| 306 The result of which can be seen in @ref{fig:voronoi}. Note that the | |
| 307 circum-circle of one of the triangles has been added to this figure, to | |
| 308 make the relationship between the Delaunay tessellation and the Voronoi | |
| 309 diagram clearer. | |
| 310 | |
| 311 @float Figure,fig:voronoi | |
| 312 @image{voronoi,8cm} | |
| 313 @caption{Delaunay triangulation and Voronoi diagram of a random set of points} | |
| 314 @end float | |
| 315 @end ifnotinfo | |
| 6855 | 316 @end ifset |
| 6832 | 317 |
| 6847 | 318 Additional information about the size of the facets of a Voronoi |
| 319 diagram, and which points of a set of points is in a polygon can be had | |
| 320 with the @code{polyarea} and @code{inpolygon} functions respectively. | |
| 6832 | 321 |
| 322 @DOCSTRING(polyarea) | |
| 323 | |
| 324 An example of the use of @code{polyarea} might be | |
| 325 | |
| 326 @example | |
| 327 @group | |
| 328 rand ("state", 2); | |
| 329 x = rand (10, 1); | |
| 330 y = rand (10, 1); | |
| 331 [c, f] = voronoin ([x, y]); | |
| 332 af = zeros (size(f)); | |
| 333 for i = 1 : length (f) | |
| 334 af(i) = polyarea (c (f @{i, :@}, 1), c (f @{i, :@}, 2)); | |
| 335 endfor | |
| 336 @end group | |
| 337 @end example | |
| 338 | |
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339 Facets of the Voronoi diagram with a vertex at infinity have infinity |
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340 area. A simplified version of @code{polyarea} for rectangles is |
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341 available with @code{rectint} |
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342 |
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343 @DOCSTRING(rectint) |
| 6832 | 344 |
| 6847 | 345 @DOCSTRING(inpolygon) |
| 346 | |
| 347 An example of the use of @code{inpolygon} might be | |
| 348 | |
| 349 @example | |
| 350 @group | |
| 351 randn ("state", 2); | |
| 352 x = randn (100, 1); | |
| 353 y = randn (100, 1); | |
| 354 vx = cos (pi * [-1 : 0.1: 1]); | |
| 355 vy = sin (pi * [-1 : 0.1 : 1]); | |
| 356 in = inpolygon (x, y, vx, vy); | |
| 357 plot(vx, vy, x(in), y(in), "r+", x(!in), y(!in), "bo"); | |
| 358 axis ([-2, 2, -2, 2]); | |
| 359 @end group | |
| 360 @end example | |
| 361 | |
| 362 @ifnotinfo | |
| 363 @noindent | |
| 364 The result of which can be seen in @ref{fig:inpolygon}. | |
| 365 | |
| 366 @float Figure,fig:inpolygon | |
| 367 @image{inpolygon,8cm} | |
| 368 @caption{Demonstration of the @code{inpolygon} function to determine the | |
| 369 points inside a polygon} | |
| 370 @end float | |
| 371 @end ifnotinfo | |
| 372 | |
| 6823 | 373 @node Convex Hull |
| 374 @section Convex Hull | |
| 375 | |
| 7001 | 376 The convex hull of a set of points is the minimum convex envelope |
| 6832 | 377 containing all of the points. Octave has the functions @code{convhull} |
| 7007 | 378 and @code{convhulln} to calculate the convex hull of 2-dimensional and |
| 6832 | 379 N-dimensional sets of points. |
| 380 | |
| 6823 | 381 @DOCSTRING(convhull) |
| 382 | |
| 383 @DOCSTRING(convhulln) | |
| 384 | |
| 6832 | 385 An example of the use of @code{convhull} is |
| 6823 | 386 |
| 6832 | 387 @example |
| 388 @group | |
| 389 x = -3:0.05:3; | |
| 390 y = abs (sin (x)); | |
| 391 k = convhull (x, y); | |
| 392 plot (x(k), y(k), "r-", x, y, "b+"); | |
| 393 axis ([-3.05, 3.05, -0.05, 1.05]); | |
| 394 @end group | |
| 395 @end example | |
| 6823 | 396 |
| 6855 | 397 @ifset HAVE_QHULL |
| 6832 | 398 @ifnotinfo |
| 399 @noindent | |
| 400 The output of the above can be seen in @ref{fig:convhull}. | |
| 6823 | 401 |
| 6832 | 402 @float Figure,fig:convhull |
| 403 @image{convhull,8cm} | |
| 404 @caption{The convex hull of a simple set of points} | |
| 405 @end float | |
| 406 @end ifnotinfo | |
| 6855 | 407 @end ifset |
| 6823 | 408 |
| 409 @node Interpolation on Scattered Data | |
| 410 @section Interpolation on Scattered Data | |
| 411 | |
| 6832 | 412 An important use of the Delaunay tessellation is that it can be used to |
| 413 interpolate from scattered data to an arbitrary set of points. To do | |
| 414 this the N-simplex of the known set of points is calculated with | |
| 415 @code{delaunay}, @code{delaunay3} or @code{delaunayn}. Then the | |
| 416 simplicies in to which the desired points are found are | |
| 417 identified. Finally the vertices of the simplicies are used to | |
| 418 interpolate to the desired points. The functions that perform this | |
| 419 interpolation are @code{griddata}, @code{griddata3} and | |
| 420 @code{griddatan}. | |
| 421 | |
| 6823 | 422 @DOCSTRING(griddata) |
| 423 | |
| 424 @DOCSTRING(griddata3) | |
| 425 | |
| 426 @DOCSTRING(griddatan) | |
| 6832 | 427 |
| 428 An example of the use of the @code{griddata} function is | |
| 429 | |
| 430 @example | |
| 431 @group | |
| 432 rand("state",1); | |
| 433 x=2*rand(1000,1)-1; | |
| 434 y=2*rand(size(x))-1; | |
| 435 z=sin(2*(x.^2+y.^2)); | |
| 436 [xx,yy]=meshgrid(linspace(-1,1,32)); | |
| 437 griddata(x,y,z,xx,yy); | |
| 438 @end group | |
| 439 @end example | |
| 440 | |
| 6855 | 441 @ifset HAVE_QHULL |
| 6832 | 442 @noindent |
| 443 that interpolates from a random scattering of points, to a uniform | |
| 444 grid. | |
| 445 @ifnotinfo | |
| 446 The output of the above can be seen in @ref{fig:griddata}. | |
| 447 | |
| 448 @float Figure,fig:griddata | |
| 449 @image{griddata,8cm} | |
| 450 @caption{Interpolation from a scattered data to a regular grid} | |
| 451 @end float | |
| 452 @end ifnotinfo | |
| 6855 | 453 @end ifset |