Gauss–Legendre polynomial for the shape control of parametric curves
Hwan Pyo Moon  1  , Soo Hyun Kim  2@  , Song-Hwa Kwon  3  
1 : Department of Mathematics, Dongguk University-Seoul
2 : Applied Algebra and Optimization Research Center, Sungkyunkwan University
3 : Department of Mathematics, The Catholic University

 The Gauss–Legendre polygon was developed as the rectifying control polygon of Pythagorean hodograph (PH) curves, whose length is the same as the arc length of the corresponding PH curve. We here focus on the polynomial basis for the Gauss–Legendre polygon. As a Bezier curve is expressed as the linear combination of the Bezier control points with the coefficients given by the Bernstein polynomials, a PH curve can be expressed as the linear combination of the Gauss-Legendre control points with the coefficients determined by some polynomials, which we call the Gauss–Legendre polynomials. We show that the Gauss–Legendre polynomials can be constructed from the Lagrange interpolator defined over the roots of the Legendre polynomials. We investigate various properties of the Gauss–Legendre polynomials including symmetry, partition of unity, critical points, derivatives, and integrals. We also analyze the pros and cons of this polynomial over other polynomials.


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