Journal for Geometry and Graphics 23 (2019), No. 2, 147--156
Copyright Heldermann Verlag 2019
A Spatial Version of the Theorem of the Angle of Circumference
Georg Glaeser
Institute of Art and Technology, University of Applied Arts, Oskar-Kokoschka-Platz 2, 1010 Vienna, Austria
georg.glaeser@uni-ak.ac.at
Boris Odehnal
Institute of Art and Technology, University of Applied Arts, Oskar-Kokoschka-Platz 2, 1010 Vienna, Austria
boris.odehnal@uni-ak.ac.at
Hellmuth Stachel
Institute of Discrete Mathematics, University of Technology, Wiedner Hauptstr. 8-10/104, 1040 Vienna, Austria
stachel@dmg.tuwien.ac.at
[Abstract-pdf]
The presented spatial version of the theorem of the angle of circumference in three-dimensional Euclidean space deals with pairs of planes $(\epsilon,\phi)$ passing through two skew straight lines $e$ and $f$, respectively, such that the angle $\alpha$ enclosed by $\epsilon$ and $\phi$ is constant. It turns out that the set of intersection lines $r = \eps\cap\phi$ is a quartic ruled surface $\Phi$ with $e\cup f$ being its double curve. We analyse the properties of $\Phi$ and discuss the special cases showing up for special values of some shape parameters such as the slope of $e$ and $f$ (with respect to a fixed plane) or the angle $\alpha$.
Keywords: Ruled surface, angle of circumference, quartic ruled surface, Thaloid, isoptic surface.
MSC: 51N20; 51N35, 51M30, 14J16
[ Fulltext-pdf (936 KB)]