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annotate +grid/Curve.m @ 99:ecf77a50d4fe feature/arclen-param

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The arclength parameter function was slow because it called fzero. Now it is constructed once and for all with splines. Much better performance.
author Martin Almquist <martin.almquist@it.uu.se>
date Sun, 06 Dec 2015 15:41:42 +0100
parents b30f3d8845f4
children bc5db54f9efd
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0
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1 classdef Curve
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2 properties
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3 g
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4 gp
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5 transformation
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0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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6
0
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7 end
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8 methods
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9 %TODO:
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10 % Errors or FD if there is no derivative function added.
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3c39dd714fb6 In Curve: Added numerical FD differentiation if derivative is not supplied. Added arc length computation based on the derivative. Added arc length parameterization (but this function is very slow.). In +util: Added fzero_vec.m and integral_vec.m, which call fzero and integral but take vector arguments.
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11 % -semi-done
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12
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13 % Concatenation of curves
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14 % Subsections of curves
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15 % Stretching of curve parameter - semi-done
0
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16 % Curve to cell array of linesegments
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17
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18 % Can supply either derivative or a difference operator D1
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19 function obj = Curve(g,gp,transformation,D1)
0
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20 default_arg('gp',[]);
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21 default_arg('transformation',[]);
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3c39dd714fb6 In Curve: Added numerical FD differentiation if derivative is not supplied. Added arc length computation based on the derivative. Added arc length parameterization (but this function is very slow.). In +util: Added fzero_vec.m and integral_vec.m, which call fzero and integral but take vector arguments.
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22 default_arg('D1',[]);
0
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23 p_test = g(0);
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24 assert(all(size(p_test) == [2,1]), 'A curve parametrization must return a 2x1 vector.');
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26 if(isempty(gp) && ~isempty(D1))
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3c39dd714fb6 In Curve: Added numerical FD differentiation if derivative is not supplied. Added arc length computation based on the derivative. Added arc length parameterization (but this function is very slow.). In +util: Added fzero_vec.m and integral_vec.m, which call fzero and integral but take vector arguments.
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27 gp = grid.Curve.numerical_derivative(g,D1);
3c39dd714fb6 In Curve: Added numerical FD differentiation if derivative is not supplied. Added arc length computation based on the derivative. Added arc length parameterization (but this function is very slow.). In +util: Added fzero_vec.m and integral_vec.m, which call fzero and integral but take vector arguments.
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28 end
3c39dd714fb6 In Curve: Added numerical FD differentiation if derivative is not supplied. Added arc length computation based on the derivative. Added arc length parameterization (but this function is very slow.). In +util: Added fzero_vec.m and integral_vec.m, which call fzero and integral but take vector arguments.
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29
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30 if ~isempty(transformation)
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31 transformation.base_g = g;
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32 transformation.base_gp = gp;
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33 [g,gp] = grid.Curve.transform_g(g,gp,transformation);
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34 end
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0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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35
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36 obj.g = g;
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37 obj.gp = gp;
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38 obj.transformation = transformation;
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3c39dd714fb6 In Curve: Added numerical FD differentiation if derivative is not supplied. Added arc length computation based on the derivative. Added arc length parameterization (but this function is very slow.). In +util: Added fzero_vec.m and integral_vec.m, which call fzero and integral but take vector arguments.
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40 end
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41
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42 function n = normal(obj,t)
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43 deriv = obj.gp(t);
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44 normalization = sqrt(sum(deriv.^2,1));
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45 n = [-deriv(2,:)./normalization; deriv(1,:)./normalization];
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46 end
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47
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48
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0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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49 % Plots a curve g(t) for 0<t<1, using n points. Returns a handle h to the plotted curve.
0
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50 % h = plot_curve(g,n)
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51 function h = plot(obj,n)
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52 default_arg('n',100);
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53
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54 t = linspace(0,1,n);
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55 p = obj.g(t);
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56 h = line(p(1,:),p(2,:));
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0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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57 end
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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58
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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59 % Plots the derivative gp(t) for 0<t<1, using n points. Returns a handle h to the plotted curve.
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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60 % h = plot_curve(gp,n)
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61 function h = plot_derivative(obj,n)
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62 default_arg('n',100);
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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63
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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64 t = linspace(0,1,n);
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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65 p = obj.gp(t);
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66 h = line(p(1,:),p(2,:));
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67 end
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68
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69 function h= plot_normals(obj,l,n,m)
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70 default_arg('l',0.1);
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71 default_arg('n',10);
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72 default_arg('m',100);
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73 t_n = linspace(0,1,n);
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74
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75 normals = obj.normal(t_n)*l;
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76
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77 n0 = obj.g(t_n);
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78 n1 = n0 + normals;
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79
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80 h = line([n0(1,:); n1(1,:)],[n0(2,:); n1(2,:)]);
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81 set(h,'Color',Color.red);
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82 obj.plot(m);
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83 end
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84
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85 function h= show(obj,name)
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86 p = obj.g(1/2);
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87 n = obj.normal(1/2);
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88 p = p + n*0.1;
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89
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90 % Add arrow
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91
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92 h = text(p(1),p(2),name);
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93 h.HorizontalAlignment = 'center';
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94 h.VerticalAlignment = 'middle';
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95
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96 obj.plot();
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97 end
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98 % Shows curve with name and arrow for direction.
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99
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3c39dd714fb6 In Curve: Added numerical FD differentiation if derivative is not supplied. Added arc length computation based on the derivative. Added arc length parameterization (but this function is very slow.). In +util: Added fzero_vec.m and integral_vec.m, which call fzero and integral but take vector arguments.
Martin Almquist <martin.almquist@it.uu.se>
parents: 0
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100
98
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parents: 87
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101 % Arc length parameterization.
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parents: 87
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102 function curve = arcLengthStretch(obj,N)
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103 default_arg('N',100);
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104 assert(~isempty(obj.gp),'This curve does not yet have a derivative added.')
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Martin Almquist <martin.almquist@it.uu.se>
parents: 0
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105
98
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parents: 87
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106 % Construct arcLength function using splines
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107 tvec = linspace(0,1,N);
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108 arcVec = grid.Curve.arcLength(obj.gp,0,tvec);
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109 arcLength = grid.Curve.spline(tvec,arcVec);
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110
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ecf77a50d4fe The arclength parameter function was slow because it called fzero. Now it is constructed once and for all with splines. Much better performance.
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parents: 98
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111 % Stretch the parameter, construct function with splines
ecf77a50d4fe The arclength parameter function was slow because it called fzero. Now it is constructed once and for all with splines. Much better performance.
Martin Almquist <martin.almquist@it.uu.se>
parents: 98
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112 arcParVec = util.fzero_vec(@(s)arcLength(s) - tvec*arcLength(1),[0-10*eps,1+10*eps]);
ecf77a50d4fe The arclength parameter function was slow because it called fzero. Now it is constructed once and for all with splines. Much better performance.
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parents: 98
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113 arcPar = grid.Curve.spline(tvec,arcParVec);
98
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114
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115 % New function and derivative
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116 g_new = @(t)obj.g(arcPar(t));
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117 gp_old = obj.gp;
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118 gp_new = @(t) normalize(gp_old(arcPar(t)));
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119
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120 curve = grid.Curve(g_new,gp_new);
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121
86
3c39dd714fb6 In Curve: Added numerical FD differentiation if derivative is not supplied. Added arc length computation based on the derivative. Added arc length parameterization (but this function is very slow.). In +util: Added fzero_vec.m and integral_vec.m, which call fzero and integral but take vector arguments.
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122 end
3c39dd714fb6 In Curve: Added numerical FD differentiation if derivative is not supplied. Added arc length computation based on the derivative. Added arc length parameterization (but this function is very slow.). In +util: Added fzero_vec.m and integral_vec.m, which call fzero and integral but take vector arguments.
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123
0
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124 % how to make it work for methods without returns
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125 function p = subsref(obj,S)
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126 %Should i add error checking here?
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127 %Maybe if you want performance you fetch obj.g and then use that
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128 switch S(1).type
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129 case '()'
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130 p = obj.g(S.subs{1});
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131 % case '.'
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132
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133 % p = obj.(S.subs);
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134 otherwise
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135 p = builtin('subsref',obj,S);
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136 % error()
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137 end
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138 end
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139
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140
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141 %% TRANSFORMATION OF A CURVE
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142 function D = reverse(C)
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143 % g = C.g;
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144 % gp = C.gp;
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145 % D = grid.Curve(@(t)g(1-t),@(t)-gp(1-t));
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146 D = C.transform([],[],-1);
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147 end
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148
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149 function D = transform(C,A,b,flip)
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150 default_arg('A',[1 0; 0 1]);
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151 default_arg('b',[0; 0]);
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152 default_arg('flip',1);
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153 if isempty(C.transformation)
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154 g = C.g;
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155 gp = C.gp;
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156 transformation.A = A;
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157 transformation.b = b;
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158 transformation.flip = flip;
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159 else
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160 g = C.transformation.base_g;
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161 gp = C.transformation.base_gp;
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162 A_old = C.transformation.A;
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163 b_old = C.transformation.b;
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164 flip_old = C.transformation.flip;
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165
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166 transformation.A = A*A_old;
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167 transformation.b = A*b_old + b;
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168 transformation.flip = flip*flip_old;
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169 end
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170
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171 D = grid.Curve(g,gp,transformation);
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172
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173 end
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174
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175 function D = translate(C,a)
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176 g = C.g;
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177 gp = C.gp;
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178
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179 % function v = g_fun(t)
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180 % x = g(t);
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181 % v(1,:) = x(1,:)+a(1);
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182 % v(2,:) = x(2,:)+a(2);
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183 % end
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184
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185 % D = grid.Curve(@g_fun,gp);
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186
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187 D = C.transform([],a);
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188 end
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189
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190 function D = mirror(C, a, b)
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191 assert_size(a,[2,1]);
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192 assert_size(b,[2,1]);
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193
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194 g = C.g;
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195 gp = C.gp;
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196
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197 l = b-a;
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198 lx = l(1);
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199 ly = l(2);
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200
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201
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202 % fprintf('Singular?\n')
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203
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204 A = [lx^2-ly^2 2*lx*ly; 2*lx*ly ly^2-lx^2]/(l'*l);
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205
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206 % function v = g_fun(t)
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207 % % v = a + A*(g(t)-a)
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208 % x = g(t);
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209
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210 % ax1 = x(1,:)-a(1);
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211 % ax2 = x(2,:)-a(2);
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212 % v(1,:) = a(1)+A(1,:)*[ax1;ax2];
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213 % v(2,:) = a(2)+A(2,:)*[ax1;ax2];
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214 % end
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215
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216 % function v = gp_fun(t)
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217 % v = A*gp(t);
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218 % end
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219
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220 % D = grid.Curve(@g_fun,@gp_fun);
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221
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222 % g = A(g-a)+a = Ag - Aa + a;
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223 b = - A*a + a;
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224 D = C.transform(A,b);
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225
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226 end
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227
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228 function D = rotate(C,a,rad)
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229 assert_size(a, [2,1]);
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230 assert_size(rad, [1,1]);
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231 g = C.g;
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232 gp = C.gp;
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233
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234
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235 A = [cos(rad) -sin(rad); sin(rad) cos(rad)];
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236
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237 % function v = g_fun(t)
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238 % % v = a + A*(g(t)-a)
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239 % x = g(t);
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240
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241 % ax1 = x(1,:)-a(1);
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242 % ax2 = x(2,:)-a(2);
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243 % v(1,:) = a(1)+A(1,:)*[ax1;ax2];
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244 % v(2,:) = a(2)+A(2,:)*[ax1;ax2];
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245 % end
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246
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247 % function v = gp_fun(t)
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248 % v = A*gp(t);
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249 % end
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250
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251 % D = grid.Curve(@g_fun,@gp_fun);
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252
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253
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254 % g = A(g-a)+a = Ag - Aa + a;
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255 b = - A*a + a;
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256 D = C.transform(A,b);
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257 end
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258 end
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259
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260 methods (Static)
86
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261
87
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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262 % Length of arc from parameter t0 to t1 (which may be vectors).
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263 % Computed using derivative.
98
b30f3d8845f4 Removed arclength property. Constructor compatible again. Translation and rotation are fast now, but ti is slow.
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264 function L = arcLength(deriv,t0,t1)
87
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265 speed = @(t) sp(deriv(t));
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266
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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267 function s = sp(deriv)
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268 s = sqrt(sum(deriv.^2,1));
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269 end
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270 L = util.integral_vec(speed,t0,t1);
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271 end
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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272
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273 function gp_out = numerical_derivative(g,D1)
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parents: 0
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274 m = length(D1); L = 1; % Assume curve parameter from 0 to 1.
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parents: 0
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275 t = linspace(0,L,m);
3c39dd714fb6 In Curve: Added numerical FD differentiation if derivative is not supplied. Added arc length computation based on the derivative. Added arc length parameterization (but this function is very slow.). In +util: Added fzero_vec.m and integral_vec.m, which call fzero and integral but take vector arguments.
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parents: 0
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276 g = g(t)';
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parents: 0
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277 gp = (D1*g)';
87
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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278
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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279 gp1_fun = grid.Curve.spline(t,gp(1,:));
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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280 gp2_fun = grid.Curve.spline(t,gp(2,:));
86
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281 gp_out = @(t) [gp1_fun(t);gp2_fun(t)];
3c39dd714fb6 In Curve: Added numerical FD differentiation if derivative is not supplied. Added arc length computation based on the derivative. Added arc length parameterization (but this function is very slow.). In +util: Added fzero_vec.m and integral_vec.m, which call fzero and integral but take vector arguments.
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282
3c39dd714fb6 In Curve: Added numerical FD differentiation if derivative is not supplied. Added arc length computation based on the derivative. Added arc length parameterization (but this function is very slow.). In +util: Added fzero_vec.m and integral_vec.m, which call fzero and integral but take vector arguments.
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283 end
3c39dd714fb6 In Curve: Added numerical FD differentiation if derivative is not supplied. Added arc length computation based on the derivative. Added arc length parameterization (but this function is very slow.). In +util: Added fzero_vec.m and integral_vec.m, which call fzero and integral but take vector arguments.
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284
87
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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285 % Returns a function handle to the spline.
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286 function f = spline(tval,fval,spline_order)
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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287 default_arg('spline_order',4);
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288 [m,~] = size(tval);
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289 assert(m==1,'Need row vectors.');
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290
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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291 % make vectors longer to be safe slightly beyond edges.
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292 dt0 = tval(2)-tval(1); dt1 = tval(end)-tval(end-1);
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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293 df0 = fval(2)-fval(1); df1 = fval(end)-fval(end-1);
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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294 tval = [tval(1)-dt0, tval, tval(end)+dt1];
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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295 fval = [fval(1)-df0, fval, fval(end)+df1];
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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296
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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297 f_spline = spapi( optknt(tval,spline_order), tval, fval );
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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298 f = @(t) fnval(f_spline,t);
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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299 end
0a29a60e0b21 In Curve: Rearranged for speed. arc_length_fun is now a property of Curve. If it is not supplied, it is computed via the derivative and spline fitting. Switching to the arc length parameterization is much faster now. The new stuff can be tested with testArcLength.m (which should be deleted after that).
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300
0
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301 function obj = line(p1, p2)
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302
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303 function v = g_fun(t)
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304 v(1,:) = p1(1) + t.*(p2(1)-p1(1));
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305 v(2,:) = p1(2) + t.*(p2(2)-p1(2));
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306 end
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307 g = @g_fun;
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308
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309 obj = grid.Curve(g);
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310 end
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311
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312 function obj = circle(c,r,phi)
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313 default_arg('phi',[0; 2*pi])
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314 default_arg('c',[0; 0])
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315 default_arg('r',1)
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316
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317 function v = g_fun(t)
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318 w = phi(1)+t*(phi(2)-phi(1));
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319 v(1,:) = c(1) + r*cos(w);
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320 v(2,:) = c(2) + r*sin(w);
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321 end
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322
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323 function v = g_fun_deriv(t)
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324 w = phi(1)+t*(phi(2)-phi(1));
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325 v(1,:) = -(phi(2)-phi(1))*r*sin(w);
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326 v(2,:) = (phi(2)-phi(1))*r*cos(w);
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327 end
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328
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329 obj = grid.Curve(@g_fun,@g_fun_deriv);
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330 end
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331
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332 function obj = bezier(p0, p1, p2, p3)
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333 function v = g_fun(t)
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334 v(1,:) = (1-t).^3*p0(1) + 3*(1-t).^2.*t*p1(1) + 3*(1-t).*t.^2*p2(1) + t.^3*p3(1);
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335 v(2,:) = (1-t).^3*p0(2) + 3*(1-t).^2.*t*p1(2) + 3*(1-t).*t.^2*p2(2) + t.^3*p3(2);
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336 end
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337
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338 function v = g_fun_deriv(t)
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339 v(1,:) = 3*(1-t).^2*(p1(1)-p0(1)) + 6*(1-t).*t*(p2(1)-p1(1)) + 3*t.^2*(p3(1)-p2(1));
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340 v(2,:) = 3*(1-t).^2*(p1(2)-p0(2)) + 6*(1-t).*t*(p2(2)-p1(2)) + 3*t.^2*(p3(2)-p2(2));
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341 end
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342
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343 obj = grid.Curve(@g_fun,@g_fun_deriv);
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344 end
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345
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346
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347 function [g_out,gp_out] = transform_g(g,gp,tr)
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348 A = tr.A;
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349 b = tr.b;
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350 flip = tr.flip;
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351
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352 function v = g_fun_noflip(t)
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353 % v = A*g + b
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354 x = g(t);
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355
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356 v(1,:) = A(1,:)*x+b(1);
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357 v(2,:) = A(2,:)*x+b(2);
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358 end
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359
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360 function v = g_fun_flip(t)
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361 % v = A*g + b
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362 x = g(1-t);
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363
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364 v(1,:) = A(1,:)*x+b(1);
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365 v(2,:) = A(2,:)*x+b(2);
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366 end
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367
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368
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369 switch flip
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370 case 1
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371 g_out = @g_fun_noflip;
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372 gp_out = @(t)A*gp(t);
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373 case -1
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374 g_out = @g_fun_flip;
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375 gp_out = @(t)-A*gp(1-t);
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376 end
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377 end
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378
86
3c39dd714fb6 In Curve: Added numerical FD differentiation if derivative is not supplied. Added arc length computation based on the derivative. Added arc length parameterization (but this function is very slow.). In +util: Added fzero_vec.m and integral_vec.m, which call fzero and integral but take vector arguments.
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379 end
0
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380 end
86
3c39dd714fb6 In Curve: Added numerical FD differentiation if derivative is not supplied. Added arc length computation based on the derivative. Added arc length parameterization (but this function is very slow.). In +util: Added fzero_vec.m and integral_vec.m, which call fzero and integral but take vector arguments.
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381
3c39dd714fb6 In Curve: Added numerical FD differentiation if derivative is not supplied. Added arc length computation based on the derivative. Added arc length parameterization (but this function is very slow.). In +util: Added fzero_vec.m and integral_vec.m, which call fzero and integral but take vector arguments.
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382 function g_norm = normalize(g0)
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383 g1 = g0(1,:); g2 = g0(2,:);
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384 normalization = sqrt(sum(g0.^2,1));
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385 g_norm = [g1./normalization; g2./normalization];
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386 end
3c39dd714fb6 In Curve: Added numerical FD differentiation if derivative is not supplied. Added arc length computation based on the derivative. Added arc length parameterization (but this function is very slow.). In +util: Added fzero_vec.m and integral_vec.m, which call fzero and integral but take vector arguments.
Martin Almquist <martin.almquist@it.uu.se>
parents: 0
diff changeset
387
3c39dd714fb6 In Curve: Added numerical FD differentiation if derivative is not supplied. Added arc length computation based on the derivative. Added arc length parameterization (but this function is very slow.). In +util: Added fzero_vec.m and integral_vec.m, which call fzero and integral but take vector arguments.
Martin Almquist <martin.almquist@it.uu.se>
parents: 0
diff changeset
388
3c39dd714fb6 In Curve: Added numerical FD differentiation if derivative is not supplied. Added arc length computation based on the derivative. Added arc length parameterization (but this function is very slow.). In +util: Added fzero_vec.m and integral_vec.m, which call fzero and integral but take vector arguments.
Martin Almquist <martin.almquist@it.uu.se>
parents: 0
diff changeset
389