Characterization of Planar Cubic Alternative curve

TL;DR

This paper thoroughly analyzes the planar cubic Alternative curve, deriving conditions for various geometric features and presenting a shape diagram to facilitate intuitive understanding and efficient computation.

Contribution

It provides a comprehensive characterization of the cubic Alternative curve, including conditions for convexity, loops, cusps, and inflection points, with a shape diagram for clarity.

Findings

Derived algebraic conditions for convexity, loops, cusps, and inflection points.

Presented a shape diagram illustrating parameter constraints.

Simplified the characterization process for the cubic Alternative curve.

Abstract

In this paper, we analyze the planar cubic Alternative curve to determine the conditions for convex, loops, cusps and inflection points. Thus cubic curve is represented by linear combination of three control points and basis function that consist of two shape parameters. By using algebraic manipulation, we can determine the constraint of shape parameters and sufficient conditions are derived which ensure that the curve is a strictly convex, loops, cusps and inflection point. We conclude the result in a shape diagram of parameters. The simplicity of this form makes characterization more intuitive and efficient to compute.