Notes on the representational possibilities of projective quadrics in four dimensions

Abstract

The paper deals with hyper-quadrics in the real projective 4-space. According to [1] there exist 11 types of hypersurfaces of 2nd order, which can be represented by 'projective normal forms' with respect to a polar simplex as coordinate frame. By interpreting this frame as a Cartesian frame in the (projectively extended) Euclidean 4-space one will receive sort of Euclidean standard types of hyper-quadrics resp., hypersurfaces of 2nd order: the sphere as representative of hyper-ellipsoids, equilateral hyper-hyperboloids, and hyper-cones of revolution. It seems to be worthwhile to visualize the "typical" projective hyper-quadrics by means of descriptive geometry in the (projectively extended) Euclidean 4-space using Maurin's method [4] or the classical (skew) axonometric mapping of that 4-space into an image plane.