The extended oloid and its contacting quadrics (Q2789203)

The oloid can be defined as the convex hull of two congruent circles lying somewhat entwined in 3-space: Their planes are orthogonal and the center of each circle lies on the other circle. The oloid is obviously a developable surface. There is a family of quadrics tangent to the oloid. This paper meticulously describes these inscribed quadrics and their curves of contact. Four of these quadrics degenerate into conic sections, two of them being real (i.e., containing real points). Furthermore, the authors consider the self-polar tetrahedron whose facets are the planes of these conic sections. The regular quadrics contain one set of ruled surfaces (1-sheet hyperboloids and one hyperbolic paraboloid). The common generators of such an inscribed quadric and the oloid are computed. This interesting surface could easily be crafted by a planar pattern (development) on a sheet of cardboard. Finally, the development of the oloid is described.