Fast and Simple Methods For Computing Control Points

TL;DR

This paper introduces simple, fast, and easy-to-implement methods for computing control points of polynomial curves and surfaces, utilizing recurrence formulas and closed-form expressions, with efficiency comparable to existing algorithms.

Contribution

It provides novel recurrence and closed-form methods for control point computation directly from polynomial representations, simplifying implementation and maintaining low computational complexity.

Findings

Methods are computationally efficient with low polynomial complexity.

Recurrence formulas work for arbitrary affine frames.

Closed-form expressions are derived for specific frames.

Abstract

The purpose of this paper is to present simple and fast methods for computing control points for polynomial curves and polynomial surfaces given explicitly in terms of polynomials (written as sums of monomials). We give recurrence formulae w.r.t. arbitrary affine frames. As a corollary, it is amusing that we can also give closed-form expressions in the case of the frame (r, s) for curves, and the frame ((1, 0, 0), (0, 1, 0), (0, 0, 1) for surfaces. Our methods have the same low polynomial (time and space) complexity as the other best known algorithms, and are very easy to implement.