Gauss-Lobatto polynomial basis for curve design with end-tangent control

초록

The Gauss–Legendre (GL) control polygon was initially proposed as a shape control tool for Pythagorean hodograph curves. Recently, it has been extended to include general polynomial curves, introducing a new representation analogous to the Bézier control polygon. In contrast to the Bézier control polygon, the GL control polygon for a polynomial curve more closely resembles the curve itself, thereby facilitating a more intuitive curve design process.The GL control polygon directly manipulates the hodograph of a curve by specifying the tangent vectors at specific parameter points. However, because these parameter points are exclusively interior points within the parameter domain, controlling the end tangent vectors of a curve becomes challenging. Therefore, for general polynomial curves, we propose using the Gauss–Lobatto (GLo) control polygons to facilitate the control of end tangent vectors.First, we define the GLo polynomials and demonstrate that the polynomial curve associated with a given GLo control polygon is unique. We also show that this curve can be easily represented by combining control points with GLo polynomials as weights, forming what we term the GLo curve. Additionally, we provide several examples of GLo curves to illustrate their usability, particularly in directly controlling the end tangent vectors of the curves. © 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

키워드