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];

[Jan]

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

[Matt J]

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.

Answer 1

[Matt J]

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

[Askic V]

@Matt J can you please elaborate your code a bit?

[Matt J]

@Askic V Not without guidance. What specifically would I be elaborating on.?

[Askic V]

I didn't understand the original question, and was wondering how you come up with that code. But in the meantime the question is modified with the additional info, so my question is not actual anymore.

Thank you.

[M]

@Matt J Thanks, but can you please give me a reference for your interpretation?

[M]

@Matt J Also, what does this line mean :

assert( rank([Aeq,beq])==rank(Aeq) , 'Hyperplanes do not intersect')

[Matt J]

Is the following correct for the 6d space?

It does look correct!

Thanks, but can you please give me a reference for your interpretation?

For example,

https://en.wikipedia.org/wiki/Affine_space#Affine_coordinates

Also, what does this line mean

It tests to see whether there is a non-empty intersection Otherwise, it makes no sense to proceed.

[M]

@Matt J Thanks much, very helpful

[M]

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]

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