Conchoid surfaces of spheres

TL;DR

This paper investigates conchoid surfaces of spheres, demonstrating their rational parameterizations through relations to quadrics, and explores their geometric properties and construction methods.

Contribution

It provides explicit rational parameterizations of conchoid surfaces of spheres using pencils of quadrics, revealing their geometric features.

Findings

Conchoid surfaces of spheres admit rational parameterizations.

Explicit parameterizations are constructed via pencils of quadrics.

The surfaces exhibit notable geometric properties.

Abstract

The conchoid of a surface $F$ with respect to given fixed point $O$ is roughly speaking the surface obtained by increasing the radius function with respect to $O$ by a constant. This paper studies {\it conchoid surfaces of spheres} and shows that these surfaces admit rational parameterizations. Explicit parameterizations of these surfaces are constructed using the relations to pencils of quadrics in $\R^3$ and $\R^4$. Moreover we point to remarkable geometric properties of these surfaces and their construction.