The extended oloid and its inscribed quadrics

TL;DR

This paper explores the geometric properties of the extended oloid, focusing on inscribed quadrics, their tangential systems, and related curves, providing detailed equations and properties of these geometric entities.

Contribution

It introduces a comprehensive analysis of the inscribed quadrics of the extended oloid, including their tangential systems, touching curves, and developable surface properties, which were not previously detailed.

Findings

Derived parameter equations for touching curves and edges of regression.

Identified the self-polar tetrahedron of the tangential system.

Determined the common generating lines of the oloid and inscribed ruled surfaces.

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.