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annotate +grid/Curve.m @ 98:b30f3d8845f4 feature/arclen-param

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Removed arclength property. Constructor compatible again. Translation and rotation are fast now, but ti is slow.
author Martin Almquist <martin.almquist@it.uu.se>
date Sun, 06 Dec 2015 13:47:12 +0100
parents 0a29a60e0b21
children ecf77a50d4fe
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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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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parents: 86
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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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b30f3d8845f4 Removed arclength property. Constructor compatible again. Translation and rotation are fast now, but ti is slow.
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parents: 87
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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
98
b30f3d8845f4 Removed arclength property. Constructor compatible again. Translation and rotation are fast now, but ti is slow.
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18 % Can supply either derivative or a difference operator D1
b30f3d8845f4 Removed arclength property. Constructor compatible again. Translation and rotation are fast now, but ti is slow.
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19 function obj = Curve(g,gp,transformation,D1)
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20 default_arg('gp',[]);
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21 default_arg('transformation',[]);
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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22 default_arg('D1',[]);
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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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b30f3d8845f4 Removed arclength property. Constructor compatible again. Translation and rotation are fast now, but ti is slow.
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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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parents: 0
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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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parents: 0
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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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parents: 0
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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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parents: 86
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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.
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50 % h = plot_curve(g,n)
98
b30f3d8845f4 Removed arclength property. Constructor compatible again. Translation and rotation are fast now, but ti is slow.
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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,:));
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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parents: 86
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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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parents: 86
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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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parents: 86
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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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parents: 86
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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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parents: 86
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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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parents: 86
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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
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.
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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parents: 87
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111 % Stretch the parameter
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parents: 87
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112 arcPar = @(t) util.fzero_vec(@(s)arcLength(s) - t*arcLength(1),[0-10*eps,1+10*eps]);
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113
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parents: 87
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114 % New function and derivative
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115 g_new = @(t)obj.g(arcPar(t));
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116 gp_old = obj.gp;
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117 gp_new = @(t) normalize(gp_old(arcPar(t)));
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118
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parents: 87
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119 curve = grid.Curve(g_new,gp_new);
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parents: 87
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120
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Martin Almquist <martin.almquist@it.uu.se>
parents: 0
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121 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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122
0
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123 % how to make it work for methods without returns
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124 function p = subsref(obj,S)
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125 %Should i add error checking here?
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126 %Maybe if you want performance you fetch obj.g and then use that
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127 switch S(1).type
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128 case '()'
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129 p = obj.g(S.subs{1});
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130 % case '.'
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131
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132 % p = obj.(S.subs);
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133 otherwise
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134 p = builtin('subsref',obj,S);
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135 % error()
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136 end
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137 end
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138
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139
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140 %% TRANSFORMATION OF A CURVE
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141 function D = reverse(C)
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142 % g = C.g;
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143 % gp = C.gp;
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144 % D = grid.Curve(@(t)g(1-t),@(t)-gp(1-t));
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145 D = C.transform([],[],-1);
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146 end
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147
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148 function D = transform(C,A,b,flip)
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149 default_arg('A',[1 0; 0 1]);
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150 default_arg('b',[0; 0]);
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151 default_arg('flip',1);
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152 if isempty(C.transformation)
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153 g = C.g;
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154 gp = C.gp;
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155 transformation.A = A;
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156 transformation.b = b;
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157 transformation.flip = flip;
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158 else
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159 g = C.transformation.base_g;
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160 gp = C.transformation.base_gp;
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161 A_old = C.transformation.A;
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162 b_old = C.transformation.b;
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163 flip_old = C.transformation.flip;
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164
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165 transformation.A = A*A_old;
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166 transformation.b = A*b_old + b;
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167 transformation.flip = flip*flip_old;
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168 end
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169
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170 D = grid.Curve(g,gp,transformation);
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171
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172 end
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173
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174 function D = translate(C,a)
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175 g = C.g;
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176 gp = C.gp;
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177
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178 % function v = g_fun(t)
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179 % x = g(t);
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180 % v(1,:) = x(1,:)+a(1);
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181 % v(2,:) = x(2,:)+a(2);
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182 % end
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183
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184 % D = grid.Curve(@g_fun,gp);
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185
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186 D = C.transform([],a);
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187 end
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188
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189 function D = mirror(C, a, b)
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190 assert_size(a,[2,1]);
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191 assert_size(b,[2,1]);
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192
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193 g = C.g;
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194 gp = C.gp;
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195
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196 l = b-a;
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197 lx = l(1);
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198 ly = l(2);
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199
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200
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201 % fprintf('Singular?\n')
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202
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203 A = [lx^2-ly^2 2*lx*ly; 2*lx*ly ly^2-lx^2]/(l'*l);
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204
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205 % function v = g_fun(t)
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206 % % v = a + A*(g(t)-a)
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207 % x = g(t);
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208
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209 % ax1 = x(1,:)-a(1);
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210 % ax2 = x(2,:)-a(2);
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211 % v(1,:) = a(1)+A(1,:)*[ax1;ax2];
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212 % v(2,:) = a(2)+A(2,:)*[ax1;ax2];
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213 % end
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214
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215 % function v = gp_fun(t)
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216 % v = A*gp(t);
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217 % end
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218
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219 % D = grid.Curve(@g_fun,@gp_fun);
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220
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221 % g = A(g-a)+a = Ag - Aa + a;
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222 b = - A*a + a;
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223 D = C.transform(A,b);
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224
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225 end
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226
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227 function D = rotate(C,a,rad)
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228 assert_size(a, [2,1]);
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229 assert_size(rad, [1,1]);
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230 g = C.g;
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231 gp = C.gp;
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232
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233
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234 A = [cos(rad) -sin(rad); sin(rad) cos(rad)];
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235
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236 % function v = g_fun(t)
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237 % % v = a + A*(g(t)-a)
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238 % x = g(t);
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239
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240 % ax1 = x(1,:)-a(1);
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241 % ax2 = x(2,:)-a(2);
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242 % v(1,:) = a(1)+A(1,:)*[ax1;ax2];
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243 % v(2,:) = a(2)+A(2,:)*[ax1;ax2];
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244 % end
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245
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246 % function v = gp_fun(t)
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247 % v = A*gp(t);
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248 % end
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249
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250 % D = grid.Curve(@g_fun,@gp_fun);
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251
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252
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253 % g = A(g-a)+a = Ag - Aa + a;
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254 b = - A*a + a;
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255 D = C.transform(A,b);
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256 end
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257 end
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258
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259 methods (Static)
86
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260
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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261 % Length of arc from parameter t0 to t1 (which may be vectors).
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262 % 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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263 function L = arcLength(deriv,t0,t1)
87
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264 speed = @(t) sp(deriv(t));
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265
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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266 function s = sp(deriv)
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267 s = sqrt(sum(deriv.^2,1));
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268 end
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269 L = util.integral_vec(speed,t0,t1);
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270 end
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271
86
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272 function gp_out = numerical_derivative(g,D1)
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273 m = length(D1); L = 1; % Assume curve parameter from 0 to 1.
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274 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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275 g = g(t)';
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276 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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277
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 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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279 gp2_fun = grid.Curve.spline(t,gp(2,:));
86
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280 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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281
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 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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283
87
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284 % Returns a function handle to the spline.
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285 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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286 default_arg('spline_order',4);
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287 [m,~] = size(tval);
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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288 assert(m==1,'Need row vectors.');
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289
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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290 % make vectors longer to be safe slightly beyond edges.
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291 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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292 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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293 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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294 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).
Martin Almquist <martin.almquist@it.uu.se>
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295
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 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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297 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).
Martin Almquist <martin.almquist@it.uu.se>
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298 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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299
0
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300 function obj = line(p1, p2)
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301
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302 function v = g_fun(t)
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303 v(1,:) = p1(1) + t.*(p2(1)-p1(1));
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304 v(2,:) = p1(2) + t.*(p2(2)-p1(2));
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305 end
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306 g = @g_fun;
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307
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308 obj = grid.Curve(g);
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309 end
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310
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311 function obj = circle(c,r,phi)
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312 default_arg('phi',[0; 2*pi])
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313 default_arg('c',[0; 0])
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314 default_arg('r',1)
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315
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316 function v = g_fun(t)
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317 w = phi(1)+t*(phi(2)-phi(1));
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318 v(1,:) = c(1) + r*cos(w);
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319 v(2,:) = c(2) + r*sin(w);
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320 end
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321
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322 function v = g_fun_deriv(t)
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323 w = phi(1)+t*(phi(2)-phi(1));
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324 v(1,:) = -(phi(2)-phi(1))*r*sin(w);
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325 v(2,:) = (phi(2)-phi(1))*r*cos(w);
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326 end
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327
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328 obj = grid.Curve(@g_fun,@g_fun_deriv);
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329 end
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330
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331 function obj = bezier(p0, p1, p2, p3)
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332 function v = g_fun(t)
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333 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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334 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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335 end
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336
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337 function v = g_fun_deriv(t)
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338 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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339 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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340 end
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341
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342 obj = grid.Curve(@g_fun,@g_fun_deriv);
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343 end
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344
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345
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346 function [g_out,gp_out] = transform_g(g,gp,tr)
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347 A = tr.A;
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348 b = tr.b;
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349 flip = tr.flip;
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350
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351 function v = g_fun_noflip(t)
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352 % v = A*g + b
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353 x = g(t);
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354
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355 v(1,:) = A(1,:)*x+b(1);
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356 v(2,:) = A(2,:)*x+b(2);
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357 end
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358
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359 function v = g_fun_flip(t)
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360 % v = A*g + b
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361 x = g(1-t);
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362
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363 v(1,:) = A(1,:)*x+b(1);
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364 v(2,:) = A(2,:)*x+b(2);
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365 end
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366
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367
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368 switch flip
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369 case 1
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370 g_out = @g_fun_noflip;
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371 gp_out = @(t)A*gp(t);
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372 case -1
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373 g_out = @g_fun_flip;
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374 gp_out = @(t)-A*gp(1-t);
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375 end
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376 end
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377
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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378 end
0
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379 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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380
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 function g_norm = normalize(g0)
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382 g1 = g0(1,:); g2 = g0(2,:);
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383 normalization = sqrt(sum(g0.^2,1));
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384 g_norm = [g1./normalization; g2./normalization];
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385 end
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diff changeset
386
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