Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 42, No. 2, pp. 531-545 (2001)

A Family of Conics and Three Special Ruled Surfaces

Hans-Peter Schröcker

Institute for Architecture, University of Applied Arts Vienna; Oskar-Kokoschka-Platz 2, A-1010 Wien, Austria; e-mail:

Abstract: \font\msbm=msbm10 \def\RR{\hbox{\msbm R}} \def\MF{{\cal F}} In [R] the authors presented a family $\MF=\{c_k\mid k\in\RR\}$ of conics. The conics $c_k$ are gained by offsetting from a given conic $c_0$ with proportional distance functions $k\delta(t)$. We investigate certain properties of $\MF$ and give the correct version of a result claimed in [R]: The distance function is unique (up to a constant factor) only if $c_0$ is not a parabola.

Furthermore we deal with the surfaces that are obtained by giving each conic $c_k \in \MF$ the $z$-coordinate $\lambda k$ with a fixed real $\lambda$. We find special metric properties of these surfaces and show that they already appeared in other context. \item{[R]} Granero Rodriguez, F.; Jiménez Hernandez, F.; Doria Iriarte, J.J.: Constructing a family of conics by curvature-depending offsetting from a given conic. Comput. Aided Geom. Design 16 (1999), 793-815.

Keywords: conic, ruled surface, surface of conic sections, rational system of conics, refraction

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