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Journal for Geometry and Graphics, Vol. 1, No. 2, pp. 105 - 118 (1997)
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The Development of the Oloid

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Hans Dirnböck, Hellmuth Stachel

Institute of Mathematics, University Klagenfurt,

Universitätsstr. 65, A-9020 Klagenfurt, Austria,

Institute of Geometry, Vienna University of Technology,

Wiedner Hauptstr. 8-10/113, A-1040 Wien, Austria,

email: stachel@geometrie.tuwien.ac.at

**Abstract:** Let two unit circles $k_A, k_B$ in perpendicular planes be given such that each circle contains the center of the other. Then the convex hull of these circles is called Oloid. In the following some geometric properties of the Oloid are treated analytically. It is proved that the development of the bounding torse $\Psi$ leads to elementary functions only. Therefore it is possible to express the rolling of the Oloid on a fixed tangent plane $\tau$ explicitly. Under this staggering motion, which is related to the well-known spatial Turbula-motion, also an ellipsoid $\Phi$ of revolution inscribed in the Oloid is rolling on $\tau$. We give parameter equations of the curve of contact in $\tau$ as well as of its counterpart on $\Phi$. The surface area of the Oloid is proved to equal the area of the unit sphere. Also the volume of the Oloid is computed.

**Keywords:** Oloid, Turbula-motion, development of torses

**Classification (MSC2000):** 51N05; 53A17

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