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

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M 2023-2-6

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https://ww2.mathworks.cn/matlabcentral/answers/1907190-how-can-we-find-the-intersection-between-two-planes-in-higher-dimensions-4d-space-and-above

编辑： Torsten 2023-6-17

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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Jan 2023-2-6

What are your inputs? Du you mean 2D planes in 4D space? 4 Points? 1 Point and 2 lines in the plane?

Matt J 2023-2-6

在 MATLAB Online 中打开

It is more compact to describe the planes in equation form Aeq*x=beq. For plane 1, this would be

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];
Aeq=null([P2;P3;P4]-P1)'
Aeq = 1×4
    0.2420    0.4472    0.5843    0.6325
beq=Aeq*P1'
beq = -3.2783e-07

and similarly for plane 2.

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采纳的回答

Matt J 2023-2-6

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1907190-how-can-we-find-the-intersection-between-two-planes-in-higher-dimensions-4d-space-and-above#answer_1164585

编辑：Matt J 2023-2-6

在 MATLAB Online 中打开

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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M 2023-6-17

Hello @Matt J, could you please tell me based on what you programed these two lines in your code:

origin = pinv(Aeq)*beq

directions = null(Aeq)

Could you please elaborate them?

Thanks

Torsten 2023-6-17

编辑：Torsten 2023-6-17

The complete set of solutions of a linear systems of equations (interpreted as the intersection of the hyperplanes) is given by one solution of the inhomogeneous system (origin) + the solutions of the homogeneous system (directions).

Read the last two paragraphs (Homogeneous solution set, Relation to nonhomogeneous systems) under

https://en.wikipedia.org/wiki/System_of_linear_equations

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