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