108 research outputs found
Total Positivity of the Cubic Trigonometric Bézier Basis
No full textWithin the general framework of Quasi Extended Chebyshev space, we prove that the cubic trigonometric Bézier basis with two shape parameters λ and μ given in Han et al. (2009) forms an optimal normalized totally positive basis for λ,μ∈(-2,1]. Moreover, we show that for λ=-2 or μ=-2 the basis is not suited for curve design from the blossom point of view. In order to compute the corresponding cubic trigonometric Bézier curves stably and efficiently, we also develop a new corner cutting algorithm
C2 quadratic trigonometric polynomial curves with local bias
No full textAbstractQuadratic trigonometric polynomial curves with local bias are presented in this paper. The quadratic trigonometric polynomial curves have C2 continuity with a non-uniform knot vector and any value of the bias, while the quadratic B-spline curves have C1 continuity. The changes of a local bias parameter will only affect two curve segments. With the bias parameters, the quadratic trigonometric polynomial curves can move locally toward or against a control vertex. A quadratic trigonometric Bézier curve is also introduced as special case of the given trigonometric polynomial curves
Multi-node higher order expansions of a function
No full textAbstractBy using the values and higher derivatives of a function at the given nodes, a kind of multi-node higher order expansion of the function is presented. The error terms of the expansions are given. Particular examples are the extensions of the Taylor polynomials, Bernstein polynomials and Lagrange interpolation polynomials. The expansions are numerical approximation polynomials and very useful particular for the functions for which the higher derivatives can be obtained easily
Piecewise quartic polynomial curves with a local shape parameter
No full textAbstractPiecewise quartic polynomial curves with a local shape parameter are presented in this paper. The given blending function is an extension of the cubic uniform B-splines. The changes of a local shape parameter will only change two curve segments. With the increase of the value of a shape parameter, the curves approach a corresponding control point. The given curves possess satisfying shape-preserving properties. The given curve can also be used to interpolate locally the control points with GC2 continuity. Thus, the given curves unify the representation of the curves for interpolating and approximating the control polygon. As an application, the piecewise polynomial curves can intersect an ellipse at different knot values by choosing the value of the shape parameter. The given curve can approximate an ellipse from the both sides and can then yield a tight envelope for an ellipse. Some computing examples for curve design are given
The Trigonometric Polynomial Like Bernstein Polynomial
No full textA symmetric basis of trigonometric polynomial space is presented. Based on the basis, symmetric trigonometric polynomial approximants like Bernstein polynomials are constructed. Two kinds of nodes are given to show that the trigonometric polynomial sequence is uniformly convergent. The convergence of the derivative of the trigonometric polynomials is shown. Trigonometric quasi-interpolants of reproducing one degree of trigonometric polynomials are constructed. Some interesting properties of the trigonometric polynomials are given
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