Harmonic rational Bézier curves, p-Bézier curves and trigonometric polynomials

Sánchez-Reyes, Javier

doi:10.1016/s0167-8396(98)00031-4

Cited by 73 publications

(51 citation statements)

References 13 publications

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“…They are used for the design of curves and they are the starting point for several generalizations [1,6,7,12,13,17,[19][20][21][22][23]25,26]: in particular to higher dimensions and to B-splines. Powerful algorithms are available for both their algebraic construction and their visualization, and their basic theory (explained beautifully in Farin's book [5]) has been examined repeatedly from different new angles: see, e.g., for the "basis function" point of view, many bases are presented in new spaces other than the polynomial space [12,13,19,20,22,25,26], [1] for the "natural generalization of Bézier curves", [7] for "barycentric" and [21] for "blossoming". Although these algorithms are very effective and widely used in practice, they have a drawback: they are not able to control the shape of the corresponding curves keeping the control polygon unchanged.…”

Section: Introductionmentioning

confidence: 99%

A novel generalization of Bézier curve and surface

Han

Huang

2008

Journal of Computational and Applied Mathematics

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A new formulation for the representation and designing of curves and surfaces is presented. It is a novel generalization of Bézier curves and surfaces. Firstly, a class of polynomial basis functions with n adjustable shape parameters is present. It is a natural extension to classical Bernstein basis functions. The corresponding Bézier curves and surfaces, the so-called Quasi-Bézier (i.e., Q-Bézier, for short) curves and surfaces, are also constructed and their properties studied. It has been shown that the main advantage compared to the ordinary Bézier curves and surfaces is that after inputting a set of control points and values of newly introduced n shape parameters, the desired curve or surface can be flexibly chosen from a set of curves or surfaces which differ either locally or globally by suitably modifying the values of the shape parameters, when the control polygon is maintained. The Q-Bézier curve and surface inherit the most properties of Bézier curve and surface and can be more approximated to the control polygon. It is visible that the properties of end-points on Q-Bézier curve and surface can be locally controlled by these shape parameters. Some examples are given by figures.

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Section: Introductionmentioning

confidence: 99%

A novel generalization of Bézier curve and surface

Han

Huang

2008

Journal of Computational and Applied Mathematics

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“…Note that, since 0 < α < π, then for any choice of α in that range the µ i are strictly positive values. Moreover, as pointed out in [17], the basis functionsB 2m i , i = 0, ..., 2m possess the following properties:…”

Section: Introduction and Motivationsmentioning

confidence: 99%

“…the normalized B-basis proposed in [17] for such a space. Note that, since 0 < α < π, then for any choice of α in that range the µ i are strictly positive values.…”

Section: Introduction and Motivationsmentioning

confidence: 99%

Algebraic-Trigonometric Pythagorean-Hodograph curves and their use for Hermite interpolation

Romani

Saini²,

Albrecht³

2014

Adv Comput Math

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In this article we define a new class of Pythagorean-Hodograph curves built-upon a sixdimensional mixed algebraic-trigonometric space, we show their fundamental properties and compare them with their well-known quintic polynomial counterpart. A complex representation for these curves is introduced and constructive approaches are provided to solve different application problems, such as interpolating C 1 Hermite data and constructing spirals as G 2 transition elements between a line segment and a circle, as well as between a pair of external circles.

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“…They include the p-Bézier [2], the quasi Bézier [3], the C-Bézier [4,5], the AH-Bézier [6], the AHT-Bézier [7], the Bézier-like models using complex numbers [8][9][10], and so on.…”

Section: Introductionmentioning

confidence: 99%