Rational complex Bezier curves

TL;DR

This paper introduces rational complex Bezier curves, extending traditional CAD curves into the complex domain, enabling new transformations and potential degree reduction for improved geometric design.

Contribution

It develops a formalism for rational complex Bezier curves, incorporating complex weights and transformations, and provides criteria for curve classification and degree reduction.

Findings

Allows application of complex projective transformations like inversion

Enables degree reduction of certain curves

Provides a simple formula to identify conic curves

Abstract

In this paper we develop the formalism of rational complex Bezier curves. This framework is a simple extension of the CAD paradigm, since it describes arc of curves in terms of control polygons and weights, which are extended to complex values. One of the major advantages of this extension is that we may make use of two different groups of projective transformations. Besides the group of projective transformations of the real plane, we have the group of complex projective transformations. This allows us to apply useful transformations like the geometric inversion to curves in design. In addition to this, the use of the complex formulation allows to lower the degree of the curves in some cases. This can be checked using the resultant of two polynomials and provides a simple formula for determining whether a rational cubic curve is a conic or not. Examples of application of the formalism to classical curves are included.