Note on the pedal locus

Extract

While the paper “On the pedal locus in non-euclidean hyperspace” was in the press, Professor H. F. Baker kindly directed the writer's attention to a reference from which it appeared that the euclidean case had first been studied by Beltrami. After publication, it was discovered that the main subject of Beltrami's paper was the non-euclidean case. He proves, by analytical methods, theorems which may be stated as follows: Let A₀, A₁,…, A_n denote the vertices of a simplex, [A], in non-euclidean space of n dimensions, and P a point such that the orthogonal projections of P on the walls of [A] lie in a flat, p; then the locus of P is an (n − 1)- fold, W, of order n + 1, which is anallagmatic for the isogonal transformation q. [A]; the isogonal conjugate of P is the absolute pole of p, and the envelope, w, of p is therefore the absolute reciprocal of W. Beltrami does not note the theorem, fundamental for the geometrical treatment of the subject, that W is the Jacobian of a certain group of n + 1 point-hyperspheres. He goes on to show that in the euclidean case the locus W breaks up into an n-ic (n − 1)-fold and the flat at infinity, while the envelope w does not, in general, break up. Finally, he notes that in two dimensions there is also a special non-euclidean case, in which W breaks up into an order-conic and a line, and w into a class-conic and a point; and that the appropriate condition is fundamentally that which is necessary for the degeneration of a certain class-conic into a pair of points. “But what are the points? And what is the corresponding condition satisfied by the absolute conic? This is a question which it would be interesting to resolve.” It does not appear that the subject has been pursued further; and in the present note an attempt is made to discuss fully the analogous case qi degeneracy in n dimensions.