Quadric-surface intersection curves: Shape and structure
The authors modify \textit{J. Levin's} algorithm [Commun. ACM 19, 555-563 (1976; Zbl 0334.68050)] for the tracing of the curve of intersection of two quadrics to take advantage of the knowledge about the curve of intersection obtained from the elementary divisors and their exponents of the pencil defined by the two quadrics. [The problem was treated comprehensively by \textit{W. Killing} in his thesis [Das Flächenbüschel 2. Ordnung, Berlin (1872)]; a modern representation of Killing's classification is, e.g., \textit{O. Mayer} [Geometrie Proiectivă, Ed. Academiei, Bucureşti (1970) \S 130]. Ray tracing is really unnecessary since the curve of intersection has a parametric representation by elliptic \(\sigma\)-functions (Killing) or \(\theta\)-functions, cf. \textit{G. Loria} [Rend. Accad. Lincei, mat.-fis. cl. Rome (4)6, 179 (1890)] that is immediately given once the relative eigenvalues and eigenvectors of the pencil are computed; and the authors' method requires even knowledge of the Weierstrass decomposition of the matrix of the pencil. If the curve of intersection has a singularity, it is of genus 0 and has a rational representation, given by Killing for all his classes. A modern reworking of Killing's book would be a great service to CAD].
scientific article; zbMATH DE number 516702