Using the normalized B-bases of vector spaces of trigonometric and hyperbolic polynomials of finite order, we specify control point configurations for the exact description of the zeroth and higher order (mixed partial) derivatives of integral curves and (hybrid) multivariate surfaces determined by coordinate functions that are exclusively given either by traditional trigonometric or hyperbolic polynomials in each of their variables. Based on homogeneous

We present a method for the interpolation of a given sequence of data points with $C^n$ continuous trigonometric spline curves of order n+1 (n > 0) that are produced by blending elliptical arcs. Ready to use explicit formulas for the control points of the interpolating arcs are also provided. Each interpolating arc depends on a global parameter $alpha in (0, pi)$ that can be used for global shape modification. Associating non-negative weights with

The classical B-spline functions of order $k geq 2$ are recursively defined as a special combination of two consecutive B-spline functions of order $k-1$. At each step, this recursive definition is based, in general, on different reparametrizations of the strictly increasing identity (linear core) function $varphi\left(u\right)=u$. This paper generalizes the concept of the classical normalized B-spline functions by considering monotone increasing

We propose an evolutionary technique (a genetic algorithm) to solve heavily constrained optimization problems defined on interpolating tensor product surfaces by adjusting the parameter values associated with the data points to be interpolated. Throughout our study we assume that the functional, which operates on these types of interpolating surfaces, is described by a surface integral and fulfills the following conditions: it is not necessarily

In CAGD curves are described mostly by means of the combination of control points and basis functions. If we associate weights with basis functions and normalize them by their weighted sum, we obtain another set of basis functions that we call quotient bases. We show some common characteristics of curves defined by such quotient basis functions. Following this approach we specify the rational counterpart of the recently introduced cyclic basis, and

Based on cyclic curves/surfaces introduced in Róth et al. (2009), we specify control point configurations that result an exact description of those closed curves and surfaces the coordinate functions of which are (separable) trigonometric polynomials of finite degree. This class of curves/surfaces comprises several famous closed curves like ellipses, epi- and hypocycloids, Lissajous curves, torus knots, foliums; and surfaces such as sphere, torus

We define a cyclic basis for the vectorspace of truncated Fourier series. The basis has several nice properties, such as positivity, summing to 1, that are often required in computer aided design, and that are used by designers in order to control curves by manipulating control points. Our curves have cyclic symmetry, i.e. the control points can be cyclically arranged and the curve does not change when the control points are cyclically permuted.

By using Riemann-Finsler geometry, we study the existence of the optimal points for a general cost function involving Finsler distances. Our minimization problem provides a model for the placement of a deposit within a domain with several markets such that the total transportation cost is minimal. Several concrete examples are studied either by precise mathematical tools or by evolutionary (computer assisted) techniques.

We provide control point based necessary and sufficient conditions for (n,m) Bézier surfaces to have linear isoparametric lines.