How can we find the intersection between two planes in higher dimensions (4d space and above)?

Torsten asked: 2023-07-01 mathematics , hyperplanes , geometry , Polynomials

Discover how to find the intersection between two planes in higher dimensions (4D space and above) using MATLAB. Get expert solutions and step-by-step guidance!

Expert Answer

Prashant Kumar answered . 2025-07-29 20:25:38

In general, intersections of two hyperplanes would be expressed algebraically by a 2xN set of linear equations Aeq*x=beq. A geometric description can be made in terms of an origin vector, which gives the position of some point in the intersection space, and a set of direction vectors which span the linear space parallel to it. Example:

Aeq=[1,2,3,4;
     5,6,7,8];
beq=[5;7];
assert(  rank([Aeq,beq])==rank(Aeq)   ,  'Hyperplanes do not intersect')
origin = pinv(Aeq)*beq

origin = 4×1
   -1.0000
   -0.2500
    0.5000
    1.2500

directions = null(Aeq)

directions = 4×2
   -0.4001   -0.3741
    0.2546    0.7970
    0.6910   -0.4717
   -0.5455    0.0488

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