# Transforming plucker coordinates

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I've been using Plucker coordinates for ray/tri intersection. I'm also working on collision detection with oriented ellipsoids, and so need to rotate everything into 'ellipsoid space' and scale it by the radii of the ellipsoid. My math background is pretty limited, and I have no idea if or how such an operation could be applied to a Plucker coordinate. I suspect that the question is academic, in that the operations required to construct a new Plucker from scratch (a vector subtraction and a cross product) are likely to be <= whatever operations are required to rotate and scale an existing Plucker coordinate. But still, I am curious as to how it might be done. Does anyone know?

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I think this would work (though I've not tried it):

Make a 4x4 matrix out of the plucker coordnates like so

{     0,    L4,    L5,    L3}{   -L4,     0,   -L2,    L1}{   -L5,    L2,     0,    L0}{   -L3,   -L1,   -L0,     0}

Then if this is M and your transformation matrix is R the transformed matrix

M' = RMR-1

will have a form similar to M from which the transformed plucker coordinates can be written down.

All this takes place in four dimensional homogeneous space, so R is both a rotation and translation matrix. Such a matrix can also scale coordinates which should work - I've not used homogeneous coordinates to do this but I have used 3x3 matrices to do simliar ellipsoid->sphere scaling in collision problems and can't see why it won't work.

But this is all from the theory of plucker coordinates, homogeneous coordinates and tensor transformations, i.e. I've not tried it. As you note recreating the plucker coordinates from the transformed geometry is probably quicker.

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As you note recreating the plucker coordinates from the transformed geometry is certainly quicker.

Be more confident John ! :)

Still it would be interesting to develop more the computation manually. Then maybe the number of ops could compete with 2 points (A and B) transfos and one segment (A,B) -> Plucker transfo.

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If transforming the plucker directly obviated the need for transforming the verts of a tri, I can imagine an optimized plucker transform might compete efficiency-wise. But if you need the transformed verts for any other purpose (sphere/edge collision detection, for example), just recreating the plucker from scratch would probably be faster.

I certainly don't have the math skills to pursue a plucker transform function. But I'd certainly be interested in how it might be done, if someone were to figure it out.

Thanks for the responses.

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Rutin
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