Karim Belabas on Mon, 13 Mar 2023 10:20:47 +0100

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Re: Determine the mirror reflection relationship between the coordinates of two sets of pairs of points in n-dimension space.

* Hongyi Zhao [2023-03-13 04:08]:
> But in all the above steps, I can only see that there is exactly one v
> is found.

(Up to normalization, yes.)

> So, the complete solution set of this problem is still not obtained so
> far.

It is: you get a (short) list V of potential v's; for each such v you get
a (short) list C of potential c's; and we know that the full solution set
is inside { s_{v,c}, v in V, c in C }. Where solutions attached to (a.v, a.c)
for a non-zero scalar 'a' have been grouped together by our normalization.

In this particular case, a single (v,c) is found, and works. (And the
affine hyperplane it corresponds to is given by the equation <x,v> = c.)


Pr Karim Belabas, U. Bordeaux, Vice-président en charge du Numérique
Institut de Mathématiques de Bordeaux UMR 5251 - (+33) 05 40 00 29 77