Gauss–Legendre polynomial basis for the shape control of polynomial curves

Hwan Pyo Moon, Soo Hyun Kim, Song Hwa Kwon

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

The Gauss–Legendre (GL) polygon was recently introduced for the shape control of Pythagorean hodograph curves. In this paper, we consider the GL polygon of general polynomial curves. The GL polygon with n+1 control points determines a polynomial curve of degree n as a barycentric combination of the control points. We identify the weight functions of this barycentric combination and define the GL polynomials, which form a basis of the polynomial space like the Bernstein polynomial basis. We investigate various properties of the GL polynomials such as the partition of unity property, symmetry, endpoint interpolation, and the critical values in comparison with the Bernstein polynomials. We also present the definite integral and higher derivatives of the GL polynomials. We then discuss the shape control of polynomial curves using the GL polygon. We claim that the design process of high degree polynomial curves using the GL polygon is much easier and more predictable than if the curve is given in the Bernstein–Bézier form. This is supported by some neat illustrative examples.

Original languageEnglish
Article number127995
JournalApplied Mathematics and Computation
Volume451
DOIs
StatePublished - 15 Aug 2023

Keywords

  • Bernstein polynomial
  • Bézier curve
  • Gauss–Legendre polygon
  • Gauss–Legendre polynomial
  • Gauss–Legendre quadrature
  • Pythagorean hodograph curves

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