Contour curves and isophotes on rational ruled surfaces

TL;DR

This paper investigates the properties of contour curves and isophotes on rational ruled surfaces, providing formulas for their genus, characterizing surfaces with rational contours, and exploring surface reconstruction from contours.

Contribution

It introduces a genus formula for isophotes on rational ruled surfaces and characterizes surfaces with rational contours, including a class of cubic surfaces.

Findings

Contours on rational ruled surfaces are rational curves.

Isophotes generally are not rational curves, except in specific cases.

Rational contours occur only on rational ruled surfaces and certain cubic surfaces.

Abstract

The ruled surfaces, i.e., surfaces generated by one parametric set of lines, are widely used in the~field of applied geometry. An~isophote on a surface is a curve consisting of surface points whose normals form a constant angle with some fixed vector. Choosing an angle equal to $\pi/2$ we obtain a special instance of a~isophote -- the so called contour curve. While contours on rational ruled surfaces are rational curves, this is no longer true for the isophotes. Hence we will provide a formula for their genus. Moreover we will show that the only surfaces with a~rational generic contour are just rational ruled surfaces and a one particular class of cubic surfaces. In addition we will deal with the reconstruction of ruled surfaces from their contours and silhouettes.