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Transformations of n-Space which Preserve a Fixed Square-Distance

Published online by Cambridge University Press:  20 November 2018

J. A. Lester
J. A. Lester*
Affiliation:
University of Waterloo, Waterloo, Ontario
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1. Introduction. Our interest here lies in the following theorem:

THEOREM 1. Assume there is defined on Rn (n ≧ 3) a “square-distance” of the form

where (gij) is a given symmetric non-singular matrix over the reals and x = (x1, …, xn), y = (y1, …, yn) ∈ Rn. Assume further that f is a bijection ofRnwhich preserves a given fixed square-distance ρ, i.e. d(x, y) = ρ if and only if d(ƒ(x),ƒ(y)) = ρ. Then (unless ρ = 0 and (gij) is positive or negative definite) ƒ(x) = Lx + ƒ(0), where L is a linear bijection ofRnsatisfying d(Lx, Ly) = ±d(x, y) for all x, y ∈ Rn (the – sign is possible if and only if ρ = 0 and (gij) has signature 0).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 2 , 01 April 1979 , pp. 392 - 395
Copyright
Copyright © Canadian Mathematical Society 1979

References

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