Approximating offset Curves using Bezier curves with high accuracy

In this paper, a new method for the approximation of offset curves is presented using the idea of the parallel derivative curves. The best uniform approximation of degree 3 with order 6 is used to construct a method to find the approximation of the offset curves for Bezier curves. The proposed method is based on the best uniform approximation, and therefore; the proposed method for constructing the offset curves induces better outcomes than the existing methods.

The best sextic approximation of hyperbola with order twelve

In this article, the best uniform approximation for the hyperbola of degree 6 that has approximation order 12 is found. The associated error function vanishes 12 times and equioscillates 13 times. For an arc of the hyperbola, the error is bounded by 2:4 x 10-4. We explain the details of the derivation and show how to apply the method. The method is simple and this encourages and motivates people working in CG and CAD to apply it in their works.

The best uniform quadratic approximation of circular arcs with high accuracy

In this article, the issue of the best uniform approximation of circular arcs with parametrically defined polynomial curves is considered. The best uniform approximation of degree 2 to a circular arc is given in explicit form. The approximation is constructed so that the error function is the Chebyshev polynomial of degree 4; the error function equioscillates five times; the approximation order is four. For θ = π/4 arcs (quarter of a circle), the uniform error is 5.5 × 10−3. The numerical examples demonstrate the efficiency and simplicity of the approximation method as well as satisfy the properties of the best uniform approximation and yield the highest possible accuracy.

Uniform Approximation of Periodical Functions by Trigonometric Sums of Special Type

The approximation characteristics of trigonometric sums Un,pψ of special type on the class Cβ,∞ψ of (ψ,β)-differentiable (in the sense of A. I. Stepanets) periodical functions are studied. Because of agreement between parameters of approximative sums and approximated classes, the solution of Kolmogorov-Nikol’skii problem is obtained in a sufficiently general case. It is shown that in a number of important cases these sums provide higher order of approximation in comparison with Fourier sums, de la Vallée Poussin sums, and others on the class Cβ,∞ψ in the uniform metric. The range of parameters in which the sums Un,pψ give the order of the best uniform approximation on the classes Cβ,∞ψ is indicated.