%   this is a demo for parametric curves
% 
 
o = [2,2,0];   % origin will be shifted otherwise we won't see the arrows inside our ellipsoid

 arrow3([0,0,0], [1,0,0],'yellow');
 arrow3([0,0,0], [0,1,0],'yellow');
 arrow3([0,0,0], [0,0,1],'magenta');    % this plots the i,j,k, vectors
 
 xmin = -2;
 xmax = 2;       % plotting window
 
 
   %%  Example 1. Spiral tube.   Compare with the previous equation!
   %
   %   its equation is  r = <(2 +sin v)*cos u, (2+sinv) sin u, cos v>
 
 du = pi/48;  
 dv = pi/48;
 
 [u,v] = meshgrid(0:du:2*pi, 0:du:2*pi); 
   X = (2 + sin(v)).*cos(u);
   Y = (2 + sin(v)).*sin(u);
   Z = cos(v);
          surf(X,Y,Z,'FaceColor',[0.8 1 0.8]);        % surface plotting
  
  % now plot the traces
      u0 = pi;
  
   x = (2 + sin(v)).*cos(u0);
   y = (2 + sin(v)).*sin(u0);
   z = cos(v);
   
     hold on
     plot3(x,y,z,...
                'r','LineWidth',3);
 
     v0 = pi;         
     u1 = (0:du:2*pi)';
   x = (2 + sin(v0))*cos(u1);
   y = (2 + sin(v0))*sin(u1);
   z =  u1*0 + cos(v0);
   
     plot3(x,y,z,...
                'LineWidth',3);      

  %% now plot the tangent plane at u = pi, v = pi;  this is around the point <-2, 0, -1> 
  
    [X,Y] = meshgrid(-3:0.2:-1, -1:0.2:1);
    Z = X*0 - 1;
           surf(X,Y,Z,'FaceColor',[1 1 0.5]);