Introduction. In my note on A 5 curve theorem generalizing the theorem of Carnot, I introduced the notion of curve cross ratio. This extension of the ordinary cross ratio is the simplest situationally invariant (3.4) case of the generalized or hypersurface cross ratio of n + l pairs of hypersurfaces in w-space which is the subject of the following lines. The generalized cross ratio is a t the s...

We prove analogues of several well-known results concerning rational maps between quadrics for the class of so-called quasilinear p-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric methods which have been successfully applied to the study of projective homogeneous varieties over fields cannot be used. We are therefore forced to take an alternat...

This paper1 continues the study of a class of compact convex hypersurfaces in R n+1 , n ≥ 1, which are boundaries of compact convex bodies obtained by taking the intersection of (solid) confocal paraboloids of revolution. Such hypersurfaces are called reflectors. In R3 reflectors arise naturally in geometrical optics and are used in design of light reflectors and reflector antennas. They are al...

3-dimensional affine hypersurfaces admitting a pointwise SO(2)-or Z 3-symmetry Abstract In (equi-)affine differential geometry, the most important algebraic invariants are the affine (Blaschke) metric h, the affine shape operator S and the difference tensor K. A hypersurface is said to admit a point-wise symmetry if at every point there exists a linear transformation preserving the affine metri...

We prove analogues of several well-known results concerning rational morphisms between quadrics for the class of so-called quasilinear p-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric methods which have been successfully applied to the study of projective homogeneous varieties over fields cannot be used. We are therefore forced to take an alt...

Orthoptic curves for the conics are well known. It is Monge's circle ellipse and hyperbola, parabola it its directrix. These level sets of quadratic functions in plane. We consider higher dimension, known as quadric hypersurfaces. For these hypersurfaces we present study their orthoptic sets, which extend idea conics.