Mapping PointsDate: 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/ |
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