Fast Approximate Implicitization of Envelope Curves using Chebyshev Polynomials

TL;DR

This paper presents a fast, approximate method for computing the implicit representation of envelope curves from rational families, using Chebyshev polynomials to improve efficiency and numerical stability.

Contribution

It introduces an efficient approximate implicitization technique employing Chebyshev polynomials for envelope curves, reducing computational cost and enhancing numerical stability.

Findings

Method achieves faster computation times.

Improves numerical conditioning of the implicitization process.

Demonstrates effectiveness through illustrative example.

Abstract

Consider a rational family of planar rational curves in a certain region of interest. We are interested in finding an approximation to the implicit representation of the envelope. Since exact implicitization methods tend to be very costly, we employ an adaptation of approximate implicitization to envelope computation. Moreover, by utilizing an orthogonal basis in the construction process, the computational times can be shortened and the numerical condition improved. We provide an example to illustrate the performance of our approach.