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Journal for Geometry and Graphics 19 (2015), No. 2, 161--177
Copyright Heldermann Verlag 2015



The Extended Oloid and Its Contacting Quadrics

Uwe Bäsel
Faculty of Mechanical and Energy Engineering, HTWK Leipzig, Karl-Liebknecht-Str. 134, 04277 Leipzig, Germany
uwe.baesel@htwk-leipzig.de

Hans Dirnböck
Nussberg 22, 9062 Moosburg, Austria



The oloid is the convex hull of two circles with equal radius in perpendicular planes so that the center of each circle lies on the other circle. It is part of a developable surface which we call extended oloid. We determine the tangential system of all contacting quadrics Q of the extended oloid O. From this result we conclude parameter equations of the touching curve c between O and Q, and of the edge of regression of O. Properties of the curves c are investigated. The self-polar tetrahedron of the tangential system of quadrics is obtained. The common generating lines of O and any ruled surface Q are determined. Furthermore, we derive the curves which are the images of c when O is developed onto the plane.

Keywords: Oloid, developable surface, torse, tangential system of quadrics, touching curve, edge of regression, self-polar tetrahedron, ruled surface.

MSC: 53A05; 51N20, 51N15

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