On an extension of Wallace's pedal property of the circumcircle

Abstract

1. Mr J. P. Gabbatt has discussed in the most recent Part of the Proceedingsof this Society the Pedal locus of a simplex in hyperspace. It is, however, possible to regard the pedal property of the circumcircle somewhat differently and so to seek other extensions. Given a circle, any three points on it are vertices of an inscribed triangle, and the feet of the perpendiculars on the sides from any fourth point of the circle are collinear. Is there any curve in space on which an analogous property holds for any five points, viz. that the feet of the perpendiculars from any one upon the faces of the tetrahedron formed by the other four are coplanar?It will be shown that curves of order n exist in Euclidean space of n dimensions on which any n + 2 points have such a property; but that the curves cannot be real if n is odd.