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

[Hongyi Zhao <hongyi.zhao@gmail.com>]

On Mon, Mar 13, 2023 at 5:19 PM Karim Belabas
<Karim.Belabas@math.u-bordeaux.fr> wrote:
>
> * 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.)

Now, let's discuss a further question as follows:

Determine the affine transformation relationship between the
coordinates of two sets of points in n-dimensional space.

For this purpose, I've provided the following example data in
4-dimensional space:

? A=[ -x, x, x+1, x, x, x; -y, y+1/6, y, y+1, y, y; -z-t, -t, z, z,
z+1, z; t, z+t, t, t, t, t+1 ];
? B=[ -x, x+2/3, x+1, x, x, x; -y, y+1/6, y, y+1, y, y; -z-t, -t, z,
z, z+1, z; t, z+t, t, t, t, t+1 ];

They represent two sets of points in 4-dimensional space, with 6
points in each set. Furthermore, I've known that these two set of
points can be connected by an affine transformation matrix, such as
following one:

? afftran=[ -1, -2, 0, 0, 0; 1, 1, 0, 0, -1/3; 0, 0, 0, 1, 0; 0, 0, 1,
0, 0; 0, 0, 0, 0, 1 ];

Now, the question is: how to determine such matrices from scratch only
based on the above two sets of points?

> Cheers,
>
>     K.B.

Regards,
Zhao

> --
> Pr Karim Belabas, U. Bordeaux, Vice-président en charge du Numérique
> Institut de Mathématiques de Bordeaux UMR 5251 - (+33) 05 40 00 29 77
> http://www.math.u-bordeaux.fr/~kbelabas/
> `