Degree elevation from Bzier curve to C-Bzier curve with corner cutting form

The existing results of curve degree elevation mainly focus on the degree of algebraic polynomials. The paper considers the elevation of degree of the trigonometric polynomial, from a Bzier curve on the algebraic polynomial space, to a C-B′ezier curve on the algebraic and trigonometric polynomial space. The matrix of degree elevation is obtained by an operator presentation and a derivation pyramid. It possesses not a recursive presentation but a direct expression. The degree elevation process can also be represented as a corner cutting form.     

Approximation of circular arcs by Bézier curves

For the circular arc of angle 0<α<π we present the explicit form of the best GC³ quartic approximation and the best GC² quartic approximations of various types, and give the explicit form of the Hausdorff distance between the circular arc and the approximate Bézier curves for each case. We also show the existence of the GC⁴ quintic approximations to the arc, and find the explicit form of the best GC³ quintic approximation in certain constraints and their distances from the arc. All approximations we construct in this paper have the optimal order of approximation, twice of the degree of approximate Bézier curves.     

Approximation of a planar cubic Bézier spiral by circular arcs

A planar cubic Bézier curve that is a spiral, i.e., its curvature varies monotonically, does not have internal cusps, loops, and inflection points. It is suitable as a design tool for applications in which fair curves are important. Since it is polynomial, it can be conveniently incorporated in CAD systems that are based on B-splines, Bézier curves, or NURBS. When machining objects, it is desirable that as much as possible of a curved toolpath be approximated by a sequence of circular arcs rather than straight-line segments. Such an arc-spline approximation of a planar cubic Bézier spiral is presented.     

The cubic trigonometric Bézier curve with two shape parameters

A cubic trigonometric Bézier curve analogous to the cubic Bézier curve, with two shape parameters, is presented in this work. The shape of the curve can be adjusted by altering the values of shape parameters while the control polygon is kept unchanged. With the shape parameters, the cubic trigonometric Bézier curves can be made close to the cubic Bézier curves or closer to the given control polygon than the cubic Bézier curves. The ellipses can be represented exactly using cubic trigonometric Bézier curves.     

Approximation of a Continuous Curve by its Bernstein-B��zier Operator

In this paper we consider integer and rational parametric Bézier curves and study the distance between the curve and its control polygon. To measure the distance we use first and second order moduli of smoothness of vector-valued function. We consider also NURBS curves with equidistant knots. Some direct approximation theorems will be presented.     

Approximation by the Bézier variant of the MKZ-Kantorovich operators in the case α < 1

The pointwise approximation properties of the Bézier variant of the MKZ-Kantorovich operators for α ≥ 1 have been studied in [Comput. Math. Appl., 39 (2000), 1-13]. The aim of this paper is to deal with the pointwise approximation of the operators for the other case 0 < α < 1. By means of some new techniques and new inequalities we establish an estimate formula on the rate of convergence of the operators for the case 0 < α < 1. In the end we propose the q-analogue of MKZK operators.      

A geometric analysis of Renegar’s condition number,and its interplay with conic curvature

For a conic linear system of the form Ax ∈K, K a convex cone, several condition measures have been extensively studied in the last dozen years. Among these, Renegar’s condition number is arguably the most prominent for its relation to data perturbation, error bounds, problem geometry, and computational complexity of algorithms. Nonetheless, is a representation-dependent measure which is usually difficult to interpret and may lead to overly conservative bounds of computational complexity and/or geometric quantities associated with the set of feasible solutions. Herein we show that Renegar’s condition number is bounded from above and below by certain purely geometric quantities associated with A and K; furthermore our bounds highlight the role of the singular values of A and their relationship with the condition number. Moreover, by using the notion of conic curvature, we show how Renegar’s condition number can be used to provide both lower and upper bounds on the width of the set of feasible solutions. This complements the literature where only lower bounds have heretofore been developed.     

Approximation of Multivariate Functions with a Certain Mixed Smoothness by Entire Functions

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Polynomial approximation of rational Bézier curves with constraints

We present an efficient method to solve the problem of the constrained least squares approximation of the rational Bézier curve by the polynomial Bézier curve. The presented algorithm uses the dual constrained Bernstein basis polynomials, and exploits their recursive properties. Examples are given, showing the effectiveness of the algorithm.     

Approximation of Classes B p,θ r by Linear Methods and Best Approximations

We investigate problems related to the approximation by linear methods and the best approximations of the classes , 1 p in the space L .     

Approximation Solutions of Nonlinear Strongly Accretive Operator Equations by Ishikawa Iteration Procedure with Errors

Let 1＜ρ≤2,E be a real ρ-uniformly smooth Banach space and T:E→E be a continuous and strongly accretive operator.The purpose of this paper is to investigate the problem of approximating solutions to the equation Tx=f by the Ishikawa iteration procedure with errors (?) where x_0 ∈ E,{u_n},{υ_n}are bounded sequences in E and{α_n},{b_n},{c_n},{a_n~'},{b_n~'},{c_n~'} are real sequences in[0,1].Under the assumption of the condition 0＜α≤b_n c_n,An≥0, it is shown that the iterative sequence{x_n}converges strongly to the unique solution of the equation Tx=f.Furthermore,under no assumption of the condition(?)(b_n~' c_n~')=0,it is also shown that{x_n}converges strongly to the unique solution of Tx=f.     

Degenerations of rational Bézier surface with weights in the form of exponential function

Rational Bézier surface is a widely used surface fitting tool in CAD. When all the weights of a rational Bézier surface go to infinity in the form of power function, the limit of surface is the regular control surface induced by some lifting function, which is called toric degenerations of rational Bézier surfaces. In this paper, we study on the degenerations of the rational Bézier surface with weights in the exponential function and indicate the difference of our result and the work of Garc′?a-Puente et al. Through the transformation of weights in the form of exponential function and power function, the regular control surface of rational Bézier surface with weights in the exponential function is defined, which is just the limit of the surface.Compared with the power function, the exponential function approaches infinity faster, which leads to surface with the weights in the form of exponential function degenerates faster.     

Approximation of smooth convex bodies by circumscribed polytopes with respect to the surface area

Let K be a convex body with C ² boundary in the Euclidean d-space. Following the work of L. Fejes Tóth, R. Vitale, R. Schneider, P.M. Gruber, S. Glasauer and M. Ludwig, best approximation of K by polytopes of restricted number of vertices or facets is well-understood if the approximation is with respect to the volume or the mean width. In this paper we consider the circumscribed polytope P _(n) of n facets with minimal surface area, and present an asymptotic formula in terms of n for the difference of surface areas of P _(n) and K.     

Approximation of functions from B p, θ l (G) by anisotropic averages

The article considers the approximation of functions from O. V. Besov's class B _p, ^l (G) by anisotropic average functions. Proofs are given of the approximation theorem, the inverse theorem of approximation theory, and the saturation theorem associated with the choice of the averaging kernel.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 80, pp. 30–47, 1978.     

Approximation of functions of several variables by trigonometric polynomials with given number of harmonics,and estimates of ε-entropy

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Approximation of Analytic Functions with an Arbitrary Order by Generalized Baskakov–Faber Operators in Compact Sets

By using a sequence \(\lambda _{n}>0\), \(n\in \mathbb {N}\) with the property that \(\lambda _{n}\rightarrow 0\) as fast we want, in this paper we obtain the approximation order \(O(\lambda _{n})\) for a generalized Baskakov–Faber operator attached to analytic functions of exponential growth in a continuum \(G\subset \mathbb {C}\). Several concrete examples of continuums G are given for which this operator can explicitly be constructed.     

Estimates of the approximation characteristics of the classes B p,θ Ω of periodic functions of several variables with given majorant of mixed moduli of continuity

We obtain order-sharp estimates of the orthogonal projection widths of the classes B _p,θ ^Ω of periodic functions of several variables whose majorant of the mixed moduli of continuity contains both exponential and logarithmic multipliers.     

Approximation by Classical Orthogonal Polynomials with Weight in Spaces L2,γ(a,b) and Widths of Some Functional Classes

Russian Mathematics - We investigate approximations of functions of classes W2(Dγ;(a,b)), r = 2, 3, …, by classical orthogonal polynomials with a weight γ in the spaces...