The extended oloid and its inscribed quadrics

Abstract

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 inscribed quadrics $\mathcal{Q}_\lambda$ of the extended oloid $\mathcal{O}$ where $\lambda$ is the system parameter. From this result we conclude parameter equations of the touching curve $\mathcal{C}_\lambda$ between $\mathcal{O}$ and $\mathcal{Q}_\lambda$, the edge of regression $\mathcal{R}$ of $\mathcal{O}$, and the asymptotes of $\mathcal{R}$. Properties of the touching curves $\mathcal{C}_\lambda$ are investigated, including the case that $\lambda\rightarrow\pm\infty$. The self-polar tetrahedron of the tangential system $\mathcal{Q}_\lambda$ is obtained. The common generating lines of $\mathcal{O}$ and any ruled surface $\mathcal{Q}_\lambda$ are determined. Furthermore, we derive the curves which are the images of $\mathcal{C}_\lambda$ and $\mathcal{R}$ when $\mathcal{O}$ is developed onto the plane.