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Math Help - Projections from 3d to 4d

  1. #1
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    Projections from 3d to 4d

    Hi there,

    I'm doing a project where the goal is to map 4d polytopes to 3d in order to represent them graphically. In the project description it advises us to:

    Choose a unit vector n in 4-dimensional space. Then the linear map x -> x - (x.n)n projects all of 4-dimensional space orthogonally on to the 3-dimensional subspace orthogonal to n. This does not solve the problem of producing a graphic of the projected object because we need to choose an orthonormal basis in that subspace to identify it with our usual coordinated is 3-dimensional space.


    A way of doing this is to use the Gram-Schmidt process. Start with the basis n,e1,e2,e3 of 4-dim space, where e1,e2,e3 are the usual standard basis vectors. Apply G-S to get an orthonormal basis n,f1,f2,f3. Then v -> [ v.f1, v.f2, v.f3 ] gives the projection in coordinates.



    There is one difficulty here that you will need to watch out for: if the last entry of n is zero then the original "basis" is not a basis. So one must instead in such cases make a different choice of three of the four possible standard basis vectors.




    I think I understand the methodology behind this (though there is always the chance I don't) but a friend of mine doing the same thing has used v -> [ v.n, v.f2, v.f3 ] to map the projection co-ordinates (where v is a vertex of the 4d polytope); is this a viable thing to do just giving a different perspective of the projection, thus a different graphic representation or will in result in a poor representation of the 4d shape? If anyone could help clear this up for me I would be very grateful!



    Thanks
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  2. #2
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    Opalg's Avatar
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    There is nothing wrong with your friend's method. The aim of the suggested construction was to give the 3-D image of the polytope as viewed from the direction of n, whereas his method will give you the image as viewed from the direction of f1.
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  3. #3
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    Talking

    Oh yeah that makes sense, thank you!!
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