Question
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];
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Expert Answer
Prashant Kumar
PhD Expert
Answered Aug 11, 2026
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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