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Math Help - Equation of a HyperPlane

  1. #1
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    Equation of a HyperPlane

    Hi,

    Am trying to construct a pyramid tree.

    In a r dimensional space, A pyramid tree is made of 2*r pyramids.

    For example in a 2 D space, we will have four pyramids (four triangles). The triangles would all share it's vertex at origin and have the four bases as the four bounding lines of the space as shown in the below image

    Equation of a HyperPlane-pyramids_2d.png

    The same pyramid in the 3 D space would be similar as above but instead of the base being a line would be a plane and the structure itself would be a pyramid. Generally in a r dimensional space, the sides of the pyramid would be a (r-1) dimensional hyperplane.

    My question is, In case of 2-D, the line equation of the sides of the Pyramid are straight-forward. (x=y and x =-y). But how do we find the equation of the sides of the pyramid in an arbitrary dimensional space.

    Any help posted or any pointers to material I should start studying will greatly be appreciated.

    Thanks in advance,
    Sriram
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  2. #2
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    Re: Equation of a HyperPlane

    I did a bit of reading and I figured out that equation of an hyperplane can be figured using a vector normal to the hyperplane and a point lying of the hyperplane (Am just extending the 3 D planar theory). Since given that, in our case the plane passes through origin, if I can find a vector normal to the plane, say \begin{displaymath} {\vec(n)} = a1*{\vec(i1)} + a2*{\vec(i2)} + ......, + an*{\vec(in)} \end{displaymath}, then the equation of the plan would be \begin{displaymath} a1*{\vec(i1)} + a2*{\vec(i2)} + ......, + an*{\vec(in)} = 0 \end{displaymath}

    How correct or wrong am I ?
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