Symplectification of Circular Arcs and Arc Splines

TL;DR

This paper explores the symplectification of circular arcs and arc splines, introducing new geometric relationships and optimization techniques within a symplectic framework for piecewise circular curves.

Contribution

It presents a novel symplectic geometric approach to analyze and optimize circular arc splines with common endpoints or tangents.

Findings

New vectorial relationships for circular arcs derived from symplectic geometry

Arc splines form a one-parameter family optimized for various criteria

Enhanced understanding of the geometric structure of circular curves

Abstract

In this article, circular arcs are considered both individually and as elements of a piecewise circular curve. The endpoint parameterization proves to be quite advantageous here. The perspective of symplectic geometry provides new vectorial relationships for the circular arc. Curves are considered whose neighboring circular elements each have a common end point or, in addition, a common tangent. These arc splines prove to be a one-parameter curve family, whereby this parameter can be optimized with regard to various criteria.