Karim Belabas on Sat, 11 Mar 2023 16:33:46 +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. 

Hi,

  Not sure what you mean by "mirror reflection transformation". If it's
an ordinary reflection s = s_H with respect to some hyperplane H then,
let v be a unit vector orthogonal to H, we have s(x) = x - 2 <x,v> v.
So s(x) - x should be proportional to a constant vector v, which is not
the case when you check the rows of A - B; for instance when you
renormalize 

  C = A - B;
  for (i=1,#C[,1], C[i,] /= C[i,1])

the matrix C should have all its rows (roughly) equal.

(Since you write that your points are in 8-dimensional space and your
matrices are 7 x 8, I gather that you wrote the point coordinates as
rows, which is not the standard PARI convention. Anyway, considering
columns doesn't work either on you example.)

If you meant something else, please be more specific.

Cheers,

    K.B.

* Hongyi Zhao [2023-03-11 08:53]:
> Hi here,
> 
> I've the following two sets of point pairs in a given dimensional of
> space, aka, 8 here, as follows:
> 
> ? A=[-1.12014, -3.17998, -18.2911, 20.1524, -6.94559, -21.8148,
> 0.975253, 7.95121; 12.8475, -6.00645, 19.3356, 19.7475, -6.17003,
> -2.66469, 24.6888, 5.86053; -16.7258, -22.5862, 20.591, 9.54292,
> 18.5927, 21.9645, 20.892, 0.0563033; -17.2971, 5.16334, -16.3141,
> 5.35391, -10.307, -13.6297, 23.5236, 24.192; -11.0926, 6.0634,
> 19.4003, 21.6093, -1.37237, -21.7654, 4.35257, -3.1105; 11.2137,
> -20.5612, -3.41736, 19.6006, -1.13627, -6.21613, 11.5885, 15.2543;
> -13.4837, 8.02265, -11.4049, 23.3052, -19.5653, 3.19629, -10.814,
> -17.1669];
> ? B=[-19.6192, -19.8916, 21.9217, 9.26674, 22.0351, 19.6121, 19.8591,
> 1.64306; 1.39387, -5.56517, 13.6578, 22.8011, -16.2283, -11.6133,
> 8.81048, -9.95819; -0.18953, 6.13486, 25.3313, 18.5031, 9.34097,
> -13.2645, 20.0343, 13.0102; -3.4197, -6.93323, 3.31249, 18.2038,
> 16.274, -18.1138, 6.36411, 23.2793; -3.91626, -0.57597, -17.0051,
> 19.8855, -3.61887, -24.0882, -0.0230154, 9.48462; -24.6641, 18.4349,
> -6.2631, 22.238, -6.2633, -5.89393, -14.8056, -11.0355; -12.9205,
> 1.0875, -18.3268, 5.77166, -15.5141, -10.0714, 25.0861, 21.7919];
> 
> I want to determine whether they are connetted by a mirror reflection
> transformation, and if so, identify such a transformation matrix
> connecting them.
> 
> I wonder if PARI/GP can help to do the trick in this case. Any tips
> will be appreciated.
> 
> Regards,
> Zhao
> -- 
> Assoc. Prof. Hongsheng Zhao <hongyi.zhao@gmail.com>
> Theory and Simulation of Materials
> Hebei Vocational University of Technology and Engineering
> No. 473, Quannan West Street, Xindu District, Xingtai, Hebei province
> 

    K.B.
--
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/
`