Mapping Points

Date: 6/25/96 at 12:15:8
From: trigonix
Subject: Mapping Points

I'm trying to map 2D points inside a 4-point convex polygon onto 
another 4-point convex polygon (a rectangle).

Formally stated: for a point p in a source polygon as defined by 
p1,p2,p3,p4 where would that point land (= p') inside a destination 
polygon as defined by p1',p2',p3',p4' ?

Any reference or formulas appreciated!

Date: 7/29/96 at 13:34:46
From: Doctor Luis
Subject: Re: Mapping Points

What you are basically asking for is a function from one quadrilateral 
onto another quadrilateral. The function exists because the two 
quadrilaterals are homeomorphic.  

Yes, there is a map. It is, however, not unique. To illustrate this 
point, consider two squares S and S', whose vertices are

  S : (0,0) (4,0)  (4,4)  (0,4)
  S': (6,0) (10,0) (10,4) (6,4)

Clearly, a function (more specifically, a bijection) f: S -> S' , 
which maps the point S(x,y) to S'(x',y') is

  f: x' = x + 6
     y' = y

Note, however, that the function g: S -> S' defined by
  
  g: x' = 6 - x
     y' = y

also maps the points in S to S', only this time the points are mapped 
in a distinct manner.

As you can see, there is no unique map from S to S'.

So, now my question is: what kind of map do you want? The answer to
this question truly depends on the nature of the problem you have in
mind. 

Feel free to reply if you want further explanations.

-Doctor Luis,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/