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

2 min readJul 4, 2023

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];

NOTE:-

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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)

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