TY - JOUR
T1 - Gauss–Legendre polynomial basis for the shape control of polynomial curves
AU - Moon, Hwan Pyo
AU - Kim, Soo Hyun
AU - Kwon, Song Hwa
N1 - Publisher Copyright:
© 2023 Elsevier Inc.
PY - 2023/8/15
Y1 - 2023/8/15
N2 - 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.
AB - 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.
KW - Bernstein polynomial
KW - Bézier curve
KW - Gauss–Legendre polygon
KW - Gauss–Legendre polynomial
KW - Gauss–Legendre quadrature
KW - Pythagorean hodograph curves
UR - http://www.scopus.com/inward/record.url?scp=85152226238&partnerID=8YFLogxK
U2 - 10.1016/j.amc.2023.127995
DO - 10.1016/j.amc.2023.127995
M3 - Article
AN - SCOPUS:85152226238
SN - 0096-3003
VL - 451
JO - Applied Mathematics and Computation
JF - Applied Mathematics and Computation
M1 - 127995
ER -