Fundamental geometric operations for Gauss–Legendre curves via linear transformations of control edges

Gauss–Legendre (GL) polynomial curves have been introduced recently as a new alternative to the classical Bézier curves, which have been widely used in Computer Aided Geometric Design (CAGD). While Bézier curves enjoy well-known nice geometric properties and allow for algorithms such as the de Casteljau algorithm and the degree elevation algorithm, GL curves provide a different algebraic structure that allows these operations to be formulated as linear transformations of the GL control edges. In this paper, we investigate fundamental properties of GL curves and present subdivision and degree elevation algorithms as well as the conversion between the GL and Bézier control polygons. All these algorithms are expressed as linear transforms of control edges with constant coefficient matrices, which can be precomputed. This formulation leads to simple and numerically stable geometric operations for GL curves. To demonstrate the stability of these operations, we compute and analyze the operator norms of the linear transformations. We also present several examples that illustrate the effectiveness of the proposed algorithms.