15 Parametric Curves And Surfaces 15.1 Parametric Curves Designing Curves • We don’t want only polygons. • Curves are used for design. Users require a simple set of controls to allow them to edit and design curves easily. • Curves should have infinite resolution, so we can zoom in and still see a smooth curve. • We want to have a compact representation. Parametric functions are of the form x(t)...

In this paper, we present arc-length estimations for quadratic rational Bézier curves using the length of polygon and distance between both end points. Our arc-length estimations coincide with the arc-length of the quadratic rational Bézier curve exactly when the weight w is 0, 1 and∞. We show that for all w > 0 our estimations are strictly increasing with respect to w. Moreover, we find the pa...

In this paper, we present arc-length estimations for quadratic rational Bézier curves using the length of polygon and distance between both end points. Our arc-length estimations coincide with the arc-length of the quadratic rational Bézier curve exactly when the weight w is 0, 1 and∞. We show that for all w > 0 our estimations are strictly increasing with respect to w. Moreover, we find the pa...

We show that all so-called C-curves are affine images of trochoids or sine curves and use this relation to investigate the occurrence of inflection points, cusps, and loops. The results are summarized in a shape diagram of C-Bézier curves, which is useful when using C-Bézier curves for curve and surface modeling.  2003 Elsevier B.V. All rights reserved.

In this paper, we start by introducing a novel disk rational Bézier based on parallel projection, whose properties are also discussed. Then applying weighted least squares, multiobjective optimization techniques and constrained quadratic programming, we achieve multi-degree reduction of this kind of disk rational Bézier curve. The paper also gives error estimation and shows some numerical examp...

A parametric equation for a modified Bézier curve is proposed for curve fitting applications. The proposed equation contains shaping parameters to adjust the shape of the fitted curve. This flexibility of shape control is expected to produce a curve which is capable of following any sets of discrete data points. A Differential Evolution (DE) optimization based technique is proposed to find the ...

This paper presents a neural network assisted entry guidance law that is designed by applying Bézier approximation. It is shown that a fully constrained approximation of a reference trajectory can be made by using the Bézier curve. Applying this approximation, an inverse dynamic system for an entry flight is solved to generate guidance command. The guidance solution thus gotten ensures terminal...

With industrial modelling tools, topological structures and free form surfaces are often managed separately, and patches used for embedding are limited to dimension 2. A new approach is to combine topology structures of any dimension with embedding of same dimension, and use topological operations to modify the shape and the properties of surfaces and volumes. The use of Chains of map as topolo...

In this paper, a new basis, to be called C-Bézier basis, is constructed for the space Γn = span{1, t, t2, . . . , tn−2, sin t, cos t} by an integral approach. Based on this basis, we define C-Bézier curves. We then show that such basis and curves share the same properties as the Bernstein basis and the Bézier curves in polynomial spaces respectively.  2003 Elsevier Science B.V. All rights rese...

We study the relationship of transformations between Legendre and Bernstein basis. Using the relationship, we present a simple and efficient method for optimal multiple degree reductions of Bézier curves with respect to the L2-norm.  2002 Elsevier Science B.V. All rights reserved.