GL- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"> <mml:mi>k</mml:mi> </mml:math> curves: a family of polynomial curves for intuitive modelling

Bézier curves are typically used for interactively designing polynomial curves, but it is well known that this becomes increasingly unintuitive as the degree of the curve grows. Gauss–Legendre (GL) curves have been proposed as an alternative for modelling high-degree polynomial curves. They come with the advantage of preserving a close relationship between the curve and its control polygon, but are costly to evaluate in their native form. We first show how to express GL curves in the Legendre basis. Building on this and taking advantage of the three-term recurrence relation of Legendre polynomials, we explore a linear-time algorithm for evaluating GL curves. We further introduce a more general family of GL- curves, with GL curves corresponding to the case and observe that GL- curves are often remarkably similar to cubic B-spline curves. Unlike Bézier curves, the start and the end of a GL- curve are not tangent to the first and the last edge of the control polygon, respectively, and we present a strategy for restoring this property.