Torsten asked . 2023-07-01

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

How can we find the intersection between two planes in higher dimensions (4d space and above)? For example we have the following 2 planes in 4d:

Plane 1

P1 =[252716585.970010 -136769230.769231 0 0]; 
P2 =[ -136769230.769231 252716585.970010 -136769230.769231 0]; 
P3= [0 -136769230.769231 252716585.970010 -136769230.769231]; 
P4 = [0 0 -136769230.769231 126358292.985005];

Plane 2

P11= [191269260.712188 -136769230.769231 0 0]; 
P22=[ -136769230.769231 259653876.096803 -136769230.769231 0]; 
P33= [0 -136769230.769231 259653876.096803 -136769230.769231]; 
P44=[0 0 -136769230.769231 129826938.048402];

mathematics , hyperplanes , geometry , Polynomials

Expert Answer

Prashant Kumar answered . 2024-12-21 00:44:33

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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