Abstract
In this work, a family of four new trigonometric Bernstein-type basis functions with four shape parameters is constructed, which form a normalized basis with optimal total positivity. Based on the new basis functions, a kind of trigonometric Bézier-type curves with four shape parameters, analogous to the cubic Bézier curves, is constructed. With appropriate choices of control points and shape parameters, the resulting trigonometric Bézier-type curves can represent exactly any arc of an ellipse or parabola. The four shape parameters have tension control roles on adjusting the shape of resulting curves. Moreover, a new corner cutting algorithm is also proposed for calculating the trigonometric Bézier-type curves stably and efficiently.
Highlights
In computer aided geometric design and computer graphics, parametric curves and surfaces are often expressed by linearly combining control points and basis functions
If the basis functions have partition of unity, nonnegativity, and total positivity, the resulting parametric curves will possess affine invariance property, convex hull property, and variation diminishing property, which are important in curves design
The classical B-spline basis functions have been widely applied in modeling parametric curves; see [1, 2]
Summary
In computer aided geometric design and computer graphics, parametric curves and surfaces are often expressed by linearly combining control points and basis functions. The purpose of this paper is to present four new trigonometric Bernstein-type basis functions constructed in the space spanned by span{1, sin2t, (1 − sin t)α(1 − λ sin t), (1 − cos t)β(1 − μ cos t)}, which form a normalized optimal totally positive basis and include the bases given in [6,7,8, 34] as special cases. Compared with the four rational trigonometric basis functions with two denominator shape parameters constructed in the space spanned by span{1, sin2t, (1 − sin t)2/[1 + (α − 2) sin t], (1 − cos t)2/[1 + (β − 2) cos t]} (see [35]), the new constructed trigonometric Bernstein-type basis functions possess four shape parameters and have more flexibility in free-form curves shape design.
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