Homotopic curves

Here is a generalization of Ivan's example, Reflection.  Let (X1(x,y),Y1(x,y)) be a continuous mapping of the plane to itself and (f(t),g(t)) a parametric curve.  A transformed curve is the curve (X1(f(t),g(t)),Y1(f(t),g(t))).  The two curves are homotopic in the sense that the original can be continuously deformed into the second.  (The Reflection example uses X1(x,y)=y and Y1(x,y)=x).)  An interesting transformation is the "polar" function: X1(x,y)=y*cos(x) and Y1(x,y)=y*sin(x).  The attached file illustrates this; you can animate on the constant c and thus see the continuous deformation.  Enjoy.