Journal for Geometry and Graphics 22 (2018), No. 2, 149--161
Copyright Heldermann Verlag 2018
The Mean Width and Integral Geometric Properties of the Oloid
Uwe Bäsel
Faculty of Mechanical and Energy Engineering, HTWK Leipzig, Karl-Liebknecht-Str. 134, 04277 Leipzig, Germany
uwe.baesel@htwk-leipzig.de
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. We calculate the mean width of the oloid in two ways, first via the integral of mean curvature, and then directly. Using this result, the surface area and the volume of the parallel body are obtained. Furthermore, we derive the expectations of the mean width, the surface area and the volume of the intersections of a fixed oloid and a moving ball, as well as of a fixed and a moving oloid.
Keywords: Oloid, convex hull, integral of mean curvature, mean width, Steiner formula, parallel body, intrinsic volumes, principal kinematic formula.
MSC: 53A05; 52A22, 52A15, 60D05
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