Mercurial > hg > octave-nkf
annotate doc/interpreter/geometry.txi @ 16828:ddac88d32d6a
Make demos in plot m-files compatible with Matlab for running comparison script.
* scripts/plot/isosurface.m, scripts/plot/line.m, scripts/plot/mesh.m,
scripts/plot/plotyy.m, scripts/plot/printd.m, scripts/plot/semilogy.m,
scripts/plot/shrinkfaces.m, scripts/plot/stairs.m, scripts/plot/stem.m,
scripts/plot/stemleaf.m, scripts/plot/tetramesh.m: Replace double quote (")
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| author | Rik <rik@octave.org> |
|---|---|
| date | Sun, 23 Jun 2013 14:47:05 -0700 |
| parents | c3fd61c59e9c |
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| rev | line source |
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1 @c Copyright (C) 2007-2012 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 the geometry code in Octave is based on the Qhull |
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23 library@footnote{Barber, C.B., Dobkin, D.P., and Huhdanpaa, H.T., |
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24 @cite{The Quickhull Algorithm for Convex Hulls}, ACM Trans. on Mathematical |
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25 Software, 22(4):469--483, Dec 1996, @url{http://www.qhull.org}}. |
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26 Some of the documentation for Qhull, particularly for the options that |
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27 can be passed to @code{delaunay}, @code{voronoi} and @code{convhull}, |
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28 etc., is relevant to Octave users. |
| 6832 | 29 |
| 6823 | 30 @menu |
| 31 * Delaunay Triangulation:: | |
| 32 * Voronoi Diagrams:: | |
| 33 * Convex Hull:: | |
| 34 * Interpolation on Scattered Data:: | |
| 35 @end menu | |
| 36 | |
| 37 @node Delaunay Triangulation | |
| 38 @section Delaunay Triangulation | |
| 39 | |
| 6832 | 40 The Delaunay triangulation is constructed from a set of |
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41 circum-circles. These circum-circles are chosen so that there are at |
| 6832 | 42 least three of the points in the set to triangulation on the |
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43 circumference of the circum-circle. None of the points in the set of |
| 6832 | 44 points falls within any of the circum-circles. |
| 45 | |
| 46 In general there are only three points on the circumference of any | |
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47 circum-circle. However, in some cases, and in particular for the |
| 6832 | 48 case of a regular grid, 4 or more points can be on a single |
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49 circum-circle. In this case the Delaunay triangulation is not unique. |
| 6832 | 50 |
| 6823 | 51 @DOCSTRING(delaunay) |
| 52 | |
| 53 The 3- and N-dimensional extension of the Delaunay triangulation are | |
| 54 given by @code{delaunay3} and @code{delaunayn} respectively. | |
| 55 @code{delaunay3} returns a set of tetrahedra that satisfy the | |
| 56 Delaunay circum-circle criteria. Similarly, @code{delaunayn} returns the | |
| 57 N-dimensional simplex satisfying the Delaunay circum-circle criteria. | |
| 7007 | 58 The N-dimensional extension of a triangulation is called a tessellation. |
| 6823 | 59 |
| 60 @DOCSTRING(delaunay3) | |
| 61 | |
| 62 @DOCSTRING(delaunayn) | |
| 63 | |
| 6832 | 64 An example of a Delaunay triangulation of a set of points is |
| 65 | |
| 66 @example | |
| 67 @group | |
| 68 rand ("state", 2); | |
| 69 x = rand (10, 1); | |
| 70 y = rand (10, 1); | |
| 71 T = delaunay (x, y); | |
| 72 X = [ x(T(:,1)); x(T(:,2)); x(T(:,3)); x(T(:,1)) ]; | |
| 73 Y = [ y(T(:,1)); y(T(:,2)); y(T(:,3)); y(T(:,1)) ]; | |
| 74 axis ([0, 1, 0, 1]); | |
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75 plot (X, Y, "b", x, y, "r*"); |
| 6832 | 76 @end group |
| 77 @end example | |
| 78 | |
| 79 @ifnotinfo | |
| 80 @noindent | |
| 81 The result of which can be seen in @ref{fig:delaunay}. | |
| 82 | |
| 83 @float Figure,fig:delaunay | |
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84 @center @image{delaunay,4in} |
| 6832 | 85 @caption{Delaunay triangulation of a random set of points} |
| 86 @end float | |
| 87 @end ifnotinfo | |
| 88 | |
| 6823 | 89 @menu |
| 6832 | 90 * Plotting the Triangulation:: |
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91 * Identifying Points in Triangulation:: |
| 6823 | 92 @end menu |
| 93 | |
| 6832 | 94 @node Plotting the Triangulation |
| 95 @subsection Plotting the Triangulation | |
| 96 | |
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97 Octave has the functions @code{triplot}, @code{trimesh}, and @code{trisurf} |
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98 to plot the Delaunay triangulation of a 2-dimensional set of points. |
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99 @code{tetramesh} will plot the triangulation of a 3-dimensional set of points. |
| 6832 | 100 |
| 101 @DOCSTRING(triplot) | |
| 102 | |
| 103 @DOCSTRING(trimesh) | |
| 104 | |
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105 @DOCSTRING(trisurf) |
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106 |
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107 @DOCSTRING(tetramesh) |
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108 |
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109 The difference between @code{triplot}, and @code{trimesh} or @code{triplot}, |
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110 is that the former only plots the 2-dimensional triangulation itself, whereas |
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111 the second two plot the value of a function @code{f (@var{x}, @var{y})}. An |
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112 example of the use of the @code{triplot} function is |
| 6832 | 113 |
| 114 @example | |
| 115 @group | |
| 116 rand ("state", 2) | |
| 117 x = rand (20, 1); | |
| 118 y = rand (20, 1); | |
| 119 tri = delaunay (x, y); | |
| 120 triplot (tri, x, y); | |
| 121 @end group | |
| 122 @end example | |
| 123 | |
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124 @noindent |
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125 which plots the Delaunay triangulation of a set of random points in |
| 6832 | 126 2-dimensions. |
| 127 @ifnotinfo | |
| 128 The output of the above can be seen in @ref{fig:triplot}. | |
| 129 | |
| 130 @float Figure,fig:triplot | |
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131 @center @image{triplot,4in} |
| 6832 | 132 @caption{Delaunay triangulation of a random set of points} |
| 133 @end float | |
| 134 @end ifnotinfo | |
| 135 | |
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136 @node Identifying Points in Triangulation |
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137 @subsection Identifying Points in Triangulation |
| 6823 | 138 |
| 139 It is often necessary to identify whether a particular point in the | |
| 7007 | 140 N-dimensional space is within the Delaunay tessellation of a set of |
| 8480 | 141 points in this N-dimensional space, and if so which N-simplex contains |
| 7007 | 142 the point and which point in the tessellation is closest to the desired |
| 6823 | 143 point. The functions @code{tsearch} and @code{dsearch} perform this |
| 144 function in a triangulation, and @code{tsearchn} and @code{dsearchn} in | |
| 7007 | 145 an N-dimensional tessellation. |
| 6823 | 146 |
| 147 To identify whether a particular point represented by a vector @var{p} | |
| 8480 | 148 falls within one of the simplices of an N-simplex, we can write the |
| 6823 | 149 Cartesian coordinates of the point in a parametric form with respect to |
| 8480 | 150 the N-simplex. This parametric form is called the Barycentric |
| 151 Coordinates of the point. If the points defining the N-simplex are given | |
| 6823 | 152 by @code{@var{N} + 1} vectors @var{t}(@var{i},:), then the Barycentric |
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153 coordinates defining the point @var{p} are given by |
| 6823 | 154 |
| 155 @example | |
| 156 @var{p} = sum (@var{beta}(1:@var{N}+1) * @var{t}(1:@var{N}+1),:) | |
| 157 @end example | |
| 158 | |
| 159 @noindent | |
| 160 where there are @code{@var{N} + 1} values @code{@var{beta}(@var{i})} | |
| 161 that together as a vector represent the Barycentric coordinates of the | |
| 162 point @var{p}. To ensure a unique solution for the values of | |
| 163 @code{@var{beta}(@var{i})} an additional criteria of | |
| 164 | |
| 165 @example | |
| 166 sum (@var{beta}(1:@var{N}+1)) == 1 | |
| 167 @end example | |
| 168 | |
| 169 @noindent | |
| 170 is imposed, and we can therefore write the above as | |
| 171 | |
| 172 @example | |
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173 @group |
| 6823 | 174 @var{p} - @var{t}(end, :) = @var{beta}(1:end-1) * (@var{t}(1:end-1, :) |
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175 - ones (@var{N}, 1) * @var{t}(end, :) |
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176 @end group |
| 6823 | 177 @end example |
| 178 | |
| 179 @noindent | |
| 180 Solving for @var{beta} we can then write | |
| 181 | |
| 182 @example | |
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183 @group |
| 6823 | 184 @var{beta}(1:end-1) = (@var{p} - @var{t}(end, :)) / (@var{t}(1:end-1, :) |
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185 - ones (@var{N}, 1) * @var{t}(end, :)) |
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186 @var{beta}(end) = sum (@var{beta}(1:end-1)) |
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187 @end group |
| 6823 | 188 @end example |
| 189 | |
| 190 @noindent | |
| 191 which gives the formula for the conversion of the Cartesian coordinates | |
| 192 of the point @var{p} to the Barycentric coordinates @var{beta}. An | |
| 193 important property of the Barycentric coordinates is that for all points | |
| 8480 | 194 in the N-simplex |
| 6823 | 195 |
| 196 @example | |
| 197 0 <= @var{beta}(@var{i}) <= 1 | |
| 198 @end example | |
| 199 | |
| 200 @noindent | |
| 201 Therefore, the test in @code{tsearch} and @code{tsearchn} essentially | |
| 202 only needs to express each point in terms of the Barycentric coordinates | |
| 8480 | 203 of each of the simplices of the N-simplex and test the values of |
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204 @var{beta}. This is exactly the implementation used in |
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205 @code{tsearchn}. @code{tsearch} is optimized for 2-dimensions and the |
| 6823 | 206 Barycentric coordinates are not explicitly formed. |
| 207 | |
| 208 @DOCSTRING(tsearch) | |
| 209 | |
| 210 @DOCSTRING(tsearchn) | |
| 211 | |
| 212 An example of the use of @code{tsearch} can be seen with the simple | |
| 213 triangulation | |
| 214 | |
| 215 @example | |
| 216 @group | |
| 217 @var{x} = [-1; -1; 1; 1]; | |
| 218 @var{y} = [-1; 1; -1; 1]; | |
| 219 @var{tri} = [1, 2, 3; 2, 3, 1]; | |
| 220 @end group | |
| 221 @end example | |
| 222 | |
| 223 @noindent | |
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224 consisting of two triangles defined by @var{tri}. We can then identify |
| 6823 | 225 which triangle a point falls in like |
| 226 | |
| 227 @example | |
| 228 @group | |
| 229 tsearch (@var{x}, @var{y}, @var{tri}, -0.5, -0.5) | |
| 230 @result{} 1 | |
| 231 tsearch (@var{x}, @var{y}, @var{tri}, 0.5, 0.5) | |
| 232 @result{} 2 | |
| 233 @end group | |
| 234 @end example | |
| 235 | |
| 236 @noindent | |
| 237 and we can confirm that a point doesn't lie within one of the triangles like | |
| 238 | |
| 239 @example | |
| 240 @group | |
| 241 tsearch (@var{x}, @var{y}, @var{tri}, 2, 2) | |
| 242 @result{} NaN | |
| 243 @end group | |
| 244 @end example | |
| 245 | |
| 246 The @code{dsearch} and @code{dsearchn} find the closest point in a | |
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247 tessellation to the desired point. The desired point does not |
| 6823 | 248 necessarily have to be in the tessellation, and even if it the returned |
| 6832 | 249 point of the tessellation does not have to be one of the vertexes of the |
| 6823 | 250 N-simplex within which the desired point is found. |
| 251 | |
| 252 @DOCSTRING(dsearch) | |
| 253 | |
| 254 @DOCSTRING(dsearchn) | |
| 255 | |
| 256 An example of the use of @code{dsearch}, using the above values of | |
| 257 @var{x}, @var{y} and @var{tri} is | |
| 258 | |
| 259 @example | |
| 260 @group | |
| 261 dsearch (@var{x}, @var{y}, @var{tri}, -2, -2) | |
| 262 @result{} 1 | |
| 263 @end group | |
| 264 @end example | |
| 265 | |
| 266 If you wish the points that are outside the tessellation to be flagged, | |
| 267 then @code{dsearchn} can be used as | |
| 268 | |
| 269 @example | |
| 270 @group | |
| 271 dsearchn ([@var{x}, @var{y}], @var{tri}, [-2, -2], NaN) | |
| 272 @result{} NaN | |
| 273 dsearchn ([@var{x}, @var{y}], @var{tri}, [-0.5, -0.5], NaN) | |
| 274 @result{} 1 | |
| 275 @end group | |
| 276 @end example | |
| 277 | |
| 278 @noindent | |
| 279 where the point outside the tessellation are then flagged with @code{NaN}. | |
| 280 | |
| 281 @node Voronoi Diagrams | |
| 282 @section Voronoi Diagrams | |
| 283 | |
| 284 A Voronoi diagram or Voronoi tessellation of a set of points @var{s} in | |
| 285 an N-dimensional space, is the tessellation of the N-dimensional space | |
| 286 such that all points in @code{@var{v}(@var{p})}, a partitions of the | |
| 287 tessellation where @var{p} is a member of @var{s}, are closer to @var{p} | |
| 288 than any other point in @var{s}. The Voronoi diagram is related to the | |
| 6832 | 289 Delaunay triangulation of a set of points, in that the vertexes of the |
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290 Voronoi tessellation are the centers of the circum-circles of the |
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291 simplices of the Delaunay tessellation. |
| 6823 | 292 |
| 293 @DOCSTRING(voronoi) | |
| 294 | |
| 295 @DOCSTRING(voronoin) | |
| 296 | |
| 6832 | 297 An example of the use of @code{voronoi} is |
| 298 | |
| 299 @example | |
| 300 @group | |
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301 rand ("state",9); |
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302 x = rand (10,1); |
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303 y = rand (10,1); |
| 6832 | 304 tri = delaunay (x, y); |
| 305 [vx, vy] = voronoi (x, y, tri); | |
| 306 triplot (tri, x, y, "b"); | |
| 307 hold on; | |
| 308 plot (vx, vy, "r"); | |
| 309 @end group | |
| 310 @end example | |
| 311 | |
| 312 @ifnotinfo | |
| 313 @noindent | |
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314 The result of which can be seen in @ref{fig:voronoi}. Note that the |
| 6832 | 315 circum-circle of one of the triangles has been added to this figure, to |
| 316 make the relationship between the Delaunay tessellation and the Voronoi | |
| 317 diagram clearer. | |
| 318 | |
| 319 @float Figure,fig:voronoi | |
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320 @center @image{voronoi,4in} |
| 6832 | 321 @caption{Delaunay triangulation and Voronoi diagram of a random set of points} |
| 322 @end float | |
| 323 @end ifnotinfo | |
| 324 | |
| 6847 | 325 Additional information about the size of the facets of a Voronoi |
| 326 diagram, and which points of a set of points is in a polygon can be had | |
| 327 with the @code{polyarea} and @code{inpolygon} functions respectively. | |
| 6832 | 328 |
| 329 @DOCSTRING(polyarea) | |
| 330 | |
| 331 An example of the use of @code{polyarea} might be | |
| 332 | |
| 333 @example | |
| 334 @group | |
| 335 rand ("state", 2); | |
| 336 x = rand (10, 1); | |
| 337 y = rand (10, 1); | |
| 338 [c, f] = voronoin ([x, y]); | |
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339 af = zeros (size (f)); |
| 6832 | 340 for i = 1 : length (f) |
| 341 af(i) = polyarea (c (f @{i, :@}, 1), c (f @{i, :@}, 2)); | |
| 342 endfor | |
| 343 @end group | |
| 344 @end example | |
| 345 | |
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346 Facets of the Voronoi diagram with a vertex at infinity have infinity |
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347 area. A simplified version of @code{polyarea} for rectangles is |
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348 available with @code{rectint} |
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349 |
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350 @DOCSTRING(rectint) |
| 6832 | 351 |
| 6847 | 352 @DOCSTRING(inpolygon) |
| 353 | |
| 354 An example of the use of @code{inpolygon} might be | |
| 355 | |
| 356 @example | |
| 357 @group | |
| 358 randn ("state", 2); | |
| 359 x = randn (100, 1); | |
| 360 y = randn (100, 1); | |
| 361 vx = cos (pi * [-1 : 0.1: 1]); | |
| 362 vy = sin (pi * [-1 : 0.1 : 1]); | |
| 363 in = inpolygon (x, y, vx, vy); | |
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364 plot (vx, vy, x(in), y(in), "r+", x(!in), y(!in), "bo"); |
| 6847 | 365 axis ([-2, 2, -2, 2]); |
| 366 @end group | |
| 367 @end example | |
| 368 | |
| 369 @ifnotinfo | |
| 370 @noindent | |
| 371 The result of which can be seen in @ref{fig:inpolygon}. | |
| 372 | |
| 373 @float Figure,fig:inpolygon | |
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374 @center @image{inpolygon,4in} |
| 6847 | 375 @caption{Demonstration of the @code{inpolygon} function to determine the |
| 376 points inside a polygon} | |
| 377 @end float | |
| 378 @end ifnotinfo | |
| 379 | |
| 6823 | 380 @node Convex Hull |
| 381 @section Convex Hull | |
| 382 | |
| 7001 | 383 The convex hull of a set of points is the minimum convex envelope |
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384 containing all of the points. Octave has the functions @code{convhull} |
| 7007 | 385 and @code{convhulln} to calculate the convex hull of 2-dimensional and |
| 6832 | 386 N-dimensional sets of points. |
| 387 | |
| 6823 | 388 @DOCSTRING(convhull) |
| 389 | |
| 390 @DOCSTRING(convhulln) | |
| 391 | |
| 6832 | 392 An example of the use of @code{convhull} is |
| 6823 | 393 |
| 6832 | 394 @example |
| 395 @group | |
| 396 x = -3:0.05:3; | |
| 397 y = abs (sin (x)); | |
| 398 k = convhull (x, y); | |
| 399 plot (x(k), y(k), "r-", x, y, "b+"); | |
| 400 axis ([-3.05, 3.05, -0.05, 1.05]); | |
| 401 @end group | |
| 402 @end example | |
| 6823 | 403 |
| 6832 | 404 @ifnotinfo |
| 405 @noindent | |
| 406 The output of the above can be seen in @ref{fig:convhull}. | |
| 6823 | 407 |
| 6832 | 408 @float Figure,fig:convhull |
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409 @center @image{convhull,4in} |
| 6832 | 410 @caption{The convex hull of a simple set of points} |
| 411 @end float | |
| 412 @end ifnotinfo | |
| 6823 | 413 |
| 414 @node Interpolation on Scattered Data | |
| 415 @section Interpolation on Scattered Data | |
| 416 | |
| 6832 | 417 An important use of the Delaunay tessellation is that it can be used to |
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418 interpolate from scattered data to an arbitrary set of points. To do |
| 6832 | 419 this the N-simplex of the known set of points is calculated with |
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420 @code{delaunay}, @code{delaunay3} or @code{delaunayn}. Then the |
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421 simplices in to which the desired points are found are |
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422 identified. Finally the vertices of the simplices are used to |
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423 interpolate to the desired points. The functions that perform this |
| 6832 | 424 interpolation are @code{griddata}, @code{griddata3} and |
| 425 @code{griddatan}. | |
| 426 | |
| 6823 | 427 @DOCSTRING(griddata) |
| 428 | |
| 429 @DOCSTRING(griddata3) | |
| 430 | |
| 431 @DOCSTRING(griddatan) | |
| 6832 | 432 |
| 433 An example of the use of the @code{griddata} function is | |
| 434 | |
| 435 @example | |
| 436 @group | |
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437 rand ("state", 1); |
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438 x = 2*rand (1000,1) - 1; |
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439 y = 2*rand (size (x)) - 1; |
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440 z = sin (2*(x.^2+y.^2)); |
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441 [xx,yy] = meshgrid (linspace (-1,1,32)); |
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442 griddata (x,y,z,xx,yy); |
| 6832 | 443 @end group |
| 444 @end example | |
| 445 | |
| 446 @noindent | |
| 447 that interpolates from a random scattering of points, to a uniform | |
| 448 grid. | |
| 449 @ifnotinfo | |
| 450 The output of the above can be seen in @ref{fig:griddata}. | |
| 451 | |
| 452 @float Figure,fig:griddata | |
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453 @center @image{griddata,4in} |
| 6832 | 454 @caption{Interpolation from a scattered data to a regular grid} |
| 455 @end float | |
| 456 @end ifnotinfo |