The Relation Between Offset and Conchoid Constructions

TL;DR

This paper explores the algebraic relationship between offset and conchoid surface constructions, revealing a rational quadratic map linking them and providing universal parameterizations.

Contribution

It introduces a simple algebraic relation between offset and conchoid surfaces via a rational quadratic map and offers universal parameterizations for both.

Findings

A rational bijective quadratic map relates offset and conchoid surfaces.

The map preserves geometric properties and is illustrated with examples.

Universal parameterizations for offset and conchoid surfaces are provided.

Abstract

The one-sided offset surface Fd of a given surface F is, roughly speaking, obtained by shifting the tangent planes of F in direction of its oriented normal vector. The conchoid surface Gd of a given surface G is roughly speaking obtained by increasing the distance of G to a fixed reference point O by d. Whereas the offset operation is well known and implemented in most CAD-software systems, the conchoid operation is less known, although already mentioned by the ancient Greeks, and recently studied by some authors. These two operations are algebraic and create new objects from given input objects. There is a surprisingly simple relation between the offset and the conchoid operation. As derived there exists a rational bijective quadratic map which transforms a given surface F and its offset surfaces Fd to a surface G and its conchoidal surface Gd, and vice versa. Geometric properties of this map are studied and illustrated at hand of some complete examples. Furthermore rational universal parameterizations for offsets and conchoid surfaces are provided.