Well... I might be able to help but I have a couple questions to clarify the problem. Are you assuming the sides of the elastic rectangle stay straight unlike a truly realistic elastic which becomes thinner when stretched? I'm guessing you assume stretching is linear as well? I think I have an understanding of how to do this if the answer is yes to both of those :) -- Rednaxela
The answer is yes to both, though the latter is a simplifying assumption ;). --Simonton
Alright, basically what you're looking for, is a way to determine the [affine transformation matrix] (an affine transformation is a transformation that essentially fits the same constraints that this simplified case of elastic stretching has) corresponding to known distortion of 4 known points. From that matrix one can easily determine the distorted location of any other point. I'll work up a nice solution to the 2-dimensional (I assume 2-dimensional is what you're wanting) form of the problem, which should be understandable without needing a background with linear algebra, tomorrow. Tonight I am tired :) -- Rednaxela
I'm not sure this problem can be described by an affine transform. If the points can be moved in arbitrary directions by arbitrary distances, then I'd say I'm sure its not an affine transform. The points can move in this way while still allowing the stretching of the material to be linear between the points. But even if the transform isn't affine, as long as the shape is still convex it should be possible to find a solution.
Label one point O for origin. Label the other 3 A, B, C, with B being the opposite corner to the origin. The point in the stretched shape is P. We have 2 parameters u and v which describe the relative distance along the edges. P should be on the line between the points (O + u.OA) and (C + u.CB), and also on the line between the points (O + v.OC) and (A + v.AB). All we have to do now is work out and solve the vector equations to get the values of u and v. :) -- Tim Foden
Actually, yeah, this can't quite be described as an affine transform, due to the cases where one side may be stretched but another side not. The problem though is still very similar to an affine transform I believe though. In 2-dimensions, linear transforms have a 2x2 matrix that maps from [x1,y1] to [x2,y2], the 4 unknowns in the matrix can be solved for by knowing two points around an origin. Affine transformations are similar, but support translations as well, can be described as a linear transformation that maps from [x1, y1, 1] to [x2, y2], and the 6 unknowns in the 3x2 matrix describing the transform can be solved for by knowing 3 points. The transforms we're talking about here also support stretching of one side with compression of another side, and I'm 99% sure can be described as a linear transformation that maps from [x1, y1, 1, x1*y1] to [x2, y2], and the 8 unknowns in the 4x2 matrix describing the transform can be solved by knowing 4 points. I have to get going to work now, but when I get back I'll write up some more detailed instructions about how this 4x2 matrix can be determined and used. To quickly summarize though, you essentially have the formule:
q = a*x + b*y + c*x*y + d
w = e*x + f*y + g*x*y + h
where a through h are the 8 unknowns, and q and w are the new coordinate systems. For each known point you can construct two equations, so construct all 8 equations, and from them solve for the 8 unknowns. If you want to know how to solve this with matrices, which is the most optimal way for a computer to solve these systems of equations, I'll explain that when I get a chance after work :-)
-- Rednaxela
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