--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/+parametrization/Curve.m	Mon Dec 21 15:09:32 2015 +0100
@@ -0,0 +1,388 @@
+classdef Curve
+    properties
+        g
+        gp
+        transformation
+    end
+
+    methods
+        %TODO:
+        % Concatenation of curves
+        % Subsections of curves
+        % Stretching of curve parameter - done for arc length.
+        % Curve to cell array of linesegments
+
+        % Returns a curve object.
+        %   g -- curve parametrization for parameter between 0 and 1
+        %  gp -- parametrization of curve derivative
+        function obj = Curve(g,gp,transformation)
+            default_arg('gp',[]);
+            default_arg('transformation',[]);
+            p_test = g(0);
+            assert(all(size(p_test) == [2,1]), 'A curve parametrization must return a 2x1 vector.');
+
+            if ~isempty(transformation)
+                transformation.base_g = g;
+                transformation.base_gp = gp;
+                [g,gp] = grid.Curve.transform_g(g,gp,transformation);
+            end
+
+            obj.g = g;
+            obj.gp = gp;
+            obj.transformation = transformation;
+
+        end
+
+        function n = normal(obj,t)
+            assert(~isempty(obj.gp),'Curve has no derivative!');
+            deriv = obj.gp(t);
+            normalization = sqrt(sum(deriv.^2,1));
+            n = [-deriv(2,:)./normalization; deriv(1,:)./normalization];
+        end
+
+
+        % Plots a curve g(t) for 0<t<1, using n points. Returns a handle h to the plotted curve.
+        %   h = plot_curve(g,n)
+        function h = plot(obj,n)
+            default_arg('n',100);
+
+            t = linspace(0,1,n);
+            p = obj.g(t);
+            h = line(p(1,:),p(2,:));
+        end
+
+        function h= plot_normals(obj,l,n,m)
+            default_arg('l',0.1);
+            default_arg('n',10);
+            default_arg('m',100);
+            t_n = linspace(0,1,n);
+
+            normals = obj.normal(t_n)*l;
+
+            n0 = obj.g(t_n);
+            n1 = n0 + normals;
+
+            h = line([n0(1,:); n1(1,:)],[n0(2,:); n1(2,:)]);
+            set(h,'Color',Color.red);
+            obj.plot(m);
+        end
+
+        function h= show(obj,name)
+            p = obj.g(1/2);
+            n = obj.normal(1/2);
+            p = p + n*0.1;
+
+            % Add arrow
+
+            h = text(p(1),p(2),name);
+            h.HorizontalAlignment = 'center';
+            h.VerticalAlignment = 'middle';
+
+            obj.plot();
+        end
+            % Shows curve with name and arrow for direction.
+
+        % Length of arc from parameter t0 to t1 (which may be vectors).
+        % Computed using derivative.
+        function L = arcLength(obj,t0,t1)
+            assert(~isempty(obj.gp),'Curve has no derivative!');
+            speed = @(t) sqrt(sum(obj.gp(t).^2,1));
+            L =  integral_vec(speed,t0,t1);
+        end
+
+        % Creates the arc length parameterization of a curve.
+        %    N -- number of points used to approximate the arclength function
+        function curve = arcLengthParametrization(obj,N)
+            default_arg('N',100);
+            assert(~isempty(obj.gp),'Curve has no derivative!');
+
+            % Construct arcLength function using splines
+            tvec = linspace(0,1,N);
+            arcVec = obj.arcLength(0,tvec);
+            tFunc = spline(arcVec,tvec); % t as a function of arcLength
+            L = obj.arcLength(0,1);
+            arcPar = @(s) tFunc(s*L);
+
+            % New function and derivative
+            g_new = @(t)obj.g(arcPar(t));
+            gp_new = @(t) normalize(obj.gp(arcPar(t)));
+            curve = grid.Curve(g_new,gp_new);
+
+        end
+
+        % how to make it work for methods without returns
+        function p = subsref(obj,S)
+            %Should i add error checking here?
+            %Maybe if you want performance you fetch obj.g and then use that
+            switch S(1).type
+                case '()'
+                    p = obj.g(S.subs{1});
+                % case '.'
+
+                    % p = obj.(S.subs);
+                otherwise
+                    p = builtin('subsref',obj,S);
+                    % error()
+            end
+        end
+
+
+        %% TRANSFORMATION OF A CURVE
+        function D = reverse(C)
+            % g = C.g;
+            % gp = C.gp;
+            % D = grid.Curve(@(t)g(1-t),@(t)-gp(1-t));
+            D = C.transform([],[],-1);
+        end
+
+        function D = transform(C,A,b,flip)
+            default_arg('A',[1 0; 0 1]);
+            default_arg('b',[0; 0]);
+            default_arg('flip',1);
+            if isempty(C.transformation)
+                g  = C.g;
+                gp = C.gp;
+                transformation.A = A;
+                transformation.b = b;
+                transformation.flip = flip;
+            else
+                g  = C.transformation.base_g;
+                gp = C.transformation.base_gp;
+                A_old = C.transformation.A;
+                b_old = C.transformation.b;
+                flip_old = C.transformation.flip;
+
+                transformation.A = A*A_old;
+                transformation.b = A*b_old + b;
+                transformation.flip = flip*flip_old;
+            end
+
+            D = grid.Curve(g,gp,transformation);
+
+        end
+
+        function D = translate(C,a)
+            g = C.g;
+            gp = C.gp;
+
+            % function v = g_fun(t)
+            %     x = g(t);
+            %     v(1,:) = x(1,:)+a(1);
+            %     v(2,:) = x(2,:)+a(2);
+            % end
+
+            % D = grid.Curve(@g_fun,gp);
+
+            D = C.transform([],a);
+        end
+
+        function D = mirror(C, a, b)
+            assert_size(a,[2,1]);
+            assert_size(b,[2,1]);
+
+            g = C.g;
+            gp = C.gp;
+
+            l = b-a;
+            lx = l(1);
+            ly = l(2);
+
+
+            % fprintf('Singular?\n')
+
+            A = [lx^2-ly^2 2*lx*ly; 2*lx*ly ly^2-lx^2]/(l'*l);
+
+            % function v = g_fun(t)
+            %     % v = a + A*(g(t)-a)
+            %     x = g(t);
+
+            %     ax1 = x(1,:)-a(1);
+            %     ax2 = x(2,:)-a(2);
+            %     v(1,:) = a(1)+A(1,:)*[ax1;ax2];
+            %     v(2,:) = a(2)+A(2,:)*[ax1;ax2];
+            % end
+
+            % function v = gp_fun(t)
+            %     v = A*gp(t);
+            % end
+
+            % D = grid.Curve(@g_fun,@gp_fun);
+
+            % g = A(g-a)+a = Ag - Aa + a;
+            b = - A*a + a;
+            D = C.transform(A,b);
+
+        end
+
+        function D = rotate(C,a,rad)
+            assert_size(a, [2,1]);
+            assert_size(rad, [1,1]);
+            g = C.g;
+            gp = C.gp;
+
+
+            A = [cos(rad) -sin(rad); sin(rad) cos(rad)];
+
+            % function v = g_fun(t)
+            %     % v = a + A*(g(t)-a)
+            %     x = g(t);
+
+            %     ax1 = x(1,:)-a(1);
+            %     ax2 = x(2,:)-a(2);
+            %     v(1,:) = a(1)+A(1,:)*[ax1;ax2];
+            %     v(2,:) = a(2)+A(2,:)*[ax1;ax2];
+            % end
+
+            % function v = gp_fun(t)
+            %     v = A*gp(t);
+            % end
+
+            % D = grid.Curve(@g_fun,@gp_fun);
+
+
+             % g = A(g-a)+a = Ag - Aa + a;
+            b = - A*a + a;
+            D = C.transform(A,b);
+        end
+    end
+
+    methods (Static)
+
+        % Computes the derivative of g: R -> R^2 using an operator D1
+        function gp_out = numericalDerivative(g,D1)
+            m = length(D1);
+            t = linspace(0,1,m);
+            gVec = g(t)';
+            gpVec = (D1*gVec)';
+
+            gp1_fun = spline(t,gpVec(1,:));
+            gp2_fun = spline(t,gpVec(2,:));
+            gp_out = @(t) [gp1_fun(t);gp2_fun(t)];
+        end
+
+        function obj = line(p1, p2)
+
+            function v = g_fun(t)
+                v(1,:) = p1(1) + t.*(p2(1)-p1(1));
+                v(2,:) = p1(2) + t.*(p2(2)-p1(2));
+            end
+            g = @g_fun;
+
+            obj = grid.Curve(g);
+        end
+
+        function obj = circle(c,r,phi)
+            default_arg('phi',[0; 2*pi])
+            default_arg('c',[0; 0])
+            default_arg('r',1)
+
+            function v = g_fun(t)
+                w = phi(1)+t*(phi(2)-phi(1));
+                v(1,:) = c(1) + r*cos(w);
+                v(2,:) = c(2) + r*sin(w);
+            end
+
+            function v = g_fun_deriv(t)
+                w = phi(1)+t*(phi(2)-phi(1));
+                v(1,:) = -(phi(2)-phi(1))*r*sin(w);
+                v(2,:) =  (phi(2)-phi(1))*r*cos(w);
+            end
+
+            obj = grid.Curve(@g_fun,@g_fun_deriv);
+        end
+
+        function obj = bezier(p0, p1, p2, p3)
+            function v = g_fun(t)
+                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);
+                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);
+            end
+
+            function v = g_fun_deriv(t)
+                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));
+                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));
+            end
+
+            obj = grid.Curve(@g_fun,@g_fun_deriv);
+        end
+
+
+        function [g_out,gp_out] = transform_g(g,gp,tr)
+            A = tr.A;
+            b = tr.b;
+            flip = tr.flip;
+
+            function v = g_fun_noflip(t)
+                % v = A*g + b
+                x = g(t);
+
+                v(1,:) = A(1,:)*x+b(1);
+                v(2,:) = A(2,:)*x+b(2);
+            end
+
+            function v = g_fun_flip(t)
+                % v = A*g + b
+                x = g(1-t);
+
+                v(1,:) = A(1,:)*x+b(1);
+                v(2,:) = A(2,:)*x+b(2);
+            end
+
+
+            switch flip
+                case 1
+                    g_out  = @g_fun_noflip;
+                    gp_out = @(t)A*gp(t);
+                case -1
+                    g_out  = @g_fun_flip;
+                    gp_out = @(t)-A*gp(1-t);
+            end
+        end
+
+    end
+end
+
+
+
+function g_norm = normalize(g0)
+    g1 = g0(1,:);
+    g2 = g0(2,:);
+    normalization = sqrt(sum(g0.^2,1));
+    g_norm = [g1./normalization; g2./normalization];
+end
+
+function I = integral_vec(f,a,b)
+% Wrapper around the built-in function integral that
+% handles multiple limits.
+
+    Na = length(a);
+    Nb = length(b);
+    assert(Na == 1 || Nb == 1 || Na==Nb,...
+        'a and b must have same length, unless one is a scalar.');
+
+    if(Na>Nb);
+        I = zeros(size(a));
+        for i = 1:Na
+            I(i) = integral(f,a(i),b);
+        end
+    elseif(Nb>Na)
+        I = zeros(size(b));
+        for i = 1:Nb
+            I(i) = integral(f,a,b(i));
+        end
+    else
+        I = zeros(size(b));
+        for i = 1:Nb
+            I(i) = integral(f,a(i),b(i));
+        end
+    end
+end
+
+% Returns a function handle to the spline.
+function f = spline(tval,fval,spline_order)
+    default_arg('spline_order',4);
+    [m,~] = size(tval);
+    assert(m==1,'Need row vectors.');
+
+    f_spline = spapi( optknt(tval,spline_order), tval, fval );
+    f = @(t) fnval(f_spline,t);
+end