Curve and Surface Geometric Modeling via Generalized Bézier-like Model (original) (raw)
Related papers
Mathematics, 2020
The main objective of this paper is to construct the various shapes and font designing of curves and to describe the curvature by using parametric and geometric continuity constraints of generalized hybrid trigonometric Bézier (GHT-Bézier) curves. The GHT-Bernstein basis functions and Bézier curve with shape parameters are presented. The parametric and geometric continuity constraints for GHT-Bézier curves are constructed. The curvature continuity provides a guarantee of smoothness geometrically between curve segments. Furthermore, we present the curvature junction of complex figures and also compare it with the curvature of the classical Bézier curve and some other applications by using the proposed GHT-Bézier curves. This approach is one of the pivotal parts of construction, which is basically due to the existence of continuity conditions and different shape parameters that permit the curve to change easily and be more flexible without altering its control points. Therefore, by ad...
A novel generalization of Bézier curve and surface
Journal of Computational and Applied Mathematics, 2008
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.
Modified Model for Bezier Curves
2004
The Bezier curve is fundamental to a wide range of challenging and practical applications such as, computer aided geometric design, postscript font representations, generic object shape descriptions and surface representation. However, a drawback of the Bezier curve is that it only considers the global information of its control points; consequently, there is often a large gap between the curve and its control polygon, which leads to a considerable error in curve representations.
The WAT Bézier Curves and Its Applications
In this paper, a kind of quasi-cubic Bézier curves by the blending of algebraic polynomials and trigonometric polynomials using weight method is presented, named WAT Bézier curves. Here weight coefficients are also shape parameters, which are called weight parameters. The interval [0, 1] of weight parameter values can be extended to [-2,π2/(π2-6)]. The WAT Bézier curves include cubic Bézier curves and C-Bézier curves () as special cases. Unlike the existing techniques based on C-Bézier methods which can approximate the Bézier curves only from single side, the WAT Bézier curves can approximate the Bézier curve from the both sides, and the change range of shape of the curves is wider than that of C-Bézier curves. The geometric effect of the alteration of this weight parameter is discussed. Some transcendental curves can be represented by the introduced curves exactly.
Using Bezier curves in medical applications
Filomat, 2016
In this paper we survey the 3D reconstruction of an object from its 2D cross-sections has many applications in different fields of sciences such as medical physics and biomedical applications. The aim of this paper is to give not only the Bezier curves in medical applications, but also by using generating functions for the Bernstein basis functions and their identities, some combinatorial sums involving binomial coefficients are deriven. Finally, we give some comments related to the above areas.
A New Class of Quasi-Cubic Trigonometric Bezier Curve and Surfaces
2016
A kind of quasi-cubic Bézier curves by the blending of algebraic polynomials and trigonometric polynomials using weight method is presented, named WAT Bézier curves. Here weight coefficients are also shape parameters, which are called weight parameters. The interval [0, 1] of weight parameter values can be extended to [ −2, π 2 / (π2 − 6 )] and the corresponding WAT Bézier curves and surfaces are defined by the introduced base functions. The WAT Bézier curves inherit most of properties similar to those of c Bézier curves, and can be adjusted easily by using the shape parameter λ. The jointing conditions of two pieces of curves with G2 and C4 continuity are discussed. With the shape parameter chosen properly, the defined curves can express exactly any plane curves or space curves defined by parametric equation based on{1, sint, cost, sint2t, cos2t} and circular helix with high degree of accuracy without using rational form. Examples are given to illustrate that the curves and surface...
Shape analysis of cubic trigonometric Bézier curves with a shape parameter
Applied Mathematics and Computation, 2010
For the cubic trigonometric polynomial curves with a shape parameter (TB curves, for short), the effects of the shape parameter on the TB curve are made clear, the shape features of the TB curve are analyzed. The necessary and sufficient conditions are derived for these curves having single or double inflection points, a loop or a cusp, or be locally or globally convex. The results are summarized in a shape diagram of TB curves, which is useful when using TB curves for curve and surface modeling. Furthermore the influences of shape parameter on the shape diagram and the ability for adjusting the shape of the curve are shown by graph examples, respectively. Crown
On a class of Bézier-like model for shape-preserving approximation
Ferdowsi University of Mashhad, 2022
A class of Bernstein-like basis functions, equipped with a shape parameter, is presented. Employing the introduced basis functions, the corresponding curve and surface in rectangular patches are defined based on some control points. It is verified that the new curve and surface have most properties of the classical Bézier curves and surfaces. The shape parameter helps to adjust the shape of the curve and surface while the control points are fixed. We prove that the proposed Bézier-like curves can preserve monotonicity and that Bézier-like surfaces can preserve axial monotonicity. Moreover, the presented curves and surfaces preserve bound constraints implied by the original data.
Shape preserving alternatives to the rational Bézier model
2001
We discus several alternatives to the rational Bézier model, based on using curves generated by mixing polynomial and trigonometric functions, and expressing them in bases with optimal shape preserving properties (normalized B-bases). For this purpose we develop new tools for finding Bbases in general spaces. We also revisit the C-Bézier curves presented by , which coincide with the helix spline segments developed by , and are nothing else than curves expressed in the normalized B-basis of the space P 1 = span{1, t, cos t, sin t}. Such curves provide a valuable alternative to the rational Bézier model, because they can deal with both free form curves and remarkable analytical shapes, including the circle, cycloid and helix. Finally, we explore extensions of the space P 1 , by mixing algebraic and trigonometric polynomials. In particular, we show that the spaces P 2 = span{1, t, cos t, sin t, cos 2t, sin 2t}, Q = span{1, t, t 2 , cos t, sin t} and I = span{1, t, cos t, sin t, t cos t, t sin t} are also suitable for shape preserving design, and we find their normalized B-basis.
A Computational Model for q -Bernstein Quasi-Minimal Bézier Surface
Journal of Mathematics
A computational model is presented to find the q -Bernstein quasi-minimal Bézier surfaces as the extremal of Dirichlet functional, and the Bézier surfaces are used quite frequently in the literature of computer science for computer graphics and the related disciplines. The recent work [1–5] on q -Bernstein–Bézier surfaces leads the way to the new generalizations of q -Bernstein polynomial Bézier surfaces for the related Plateau–Bézier problem. The q -Bernstein polynomial-based Plateau–Bézier problem is the minimal area surface amongst all the q -Bernstein polynomial-based Bézier surfaces, spanned by the prescribed boundary. Instead of usual area functional that depends on square root of its integrand, we choose the Dirichlet functional. Related Euler–Lagrange equation is a partial differential equation, for which solutions are known for a few special cases to obtain the corresponding minimal surface. Instead of solving the partial differential equation, we can find the optimal condi...