I have two lists of ordered points in $\Bbb{R}^2$, for example in the following figure the lists are $S_1=\{1,2,3,4,5\}$ and $S_2=\{a,b,c,d,e\}$.
A list of ordered points models my idea of shape so the figure shows the shape $S_1$ and $S_2$; the border of the shape is the list of line segments connecting the points but my shape has also all the points inside it.
I would like to transform the shape $S_1$ into $S_2$ and I know that $1$ should be mapped to $a$, (write it as $1\to a$), then $2\to b$, $3\to c$, $4\to d$ and finally $5\to e$.
That is fine for the border but how can I map all the inside points? What is the destination of the generic point $(x_*,y_*)$?
$(x_*,y_*)\to ?$
I need an algorithm to solve this problem, I think it could be a well known problem but I have no idea how to search for it. I vaguely remember that Topology is related to such things but I am looking forward a pragmatic solution.
If your polygon is a triangle, the solution is simple: map a point in your source triangle to the point in your destination triangle that has the same barycentric coordinates, convenient local coordinates that take the values $(1,0,0),(0,1,0),(0,0,1)$ on the triangle vertices, and are defined on all points in the triangle, too.
If your polygon is an $n$-gon, you can always triangularize it and do the interpolation on a triangle-by-triangle basis. That is likely the simplest way, but there are other ways to generalize barycentric coordinates to $n$-gons.