
Journal for Geometry and Graphics 23 (2019), No. 2, 147156 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, OskarKokoschkaPlatz 2, 1010 Vienna, Austria georg.glaeser@uniak.ac.at Boris Odehnal Institute of Art and Technology, University of Applied Arts, OskarKokoschkaPlatz 2, 1010 Vienna, Austria boris.odehnal@uniak.ac.at Hellmuth Stachel Institute of Discrete Mathematics, University of Technology, Wiedner Hauptstr. 810/104, 1040 Vienna, Austria stachel@dmg.tuwien.ac.at [Abstractpdf] The presented spatial version of the theorem of the angle of circumference in threedimensional 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 [ Fulltextpdf (936 KB)] 