Narrow Results

Subdivision schemes generate self-similar curves and surfaces for which it has a familiar connection between fractal curves and surfaces generated by iterated function systems (IFS). Overveld [Comput.-Aided Des. 22(9) (1990) 591–597] proved that the subdivision matrices can be perturbated in such a way that it is possible to get fractal-like curves that are perturbated Bézier cubic curves. In this work, we extend the Overveld scheme to $n$th degree curves, and deduce the condition for curvature continuity and convex hull property. We find the conditions for positive preserving fractal-like Bézier curves in the proposed subdivision matrices. The resulting 2D/3D curves from these binary subdivision matrices resemble with fractal images. Finally, the dependence of the shape of these fractal-like curves on the elements of subdivision matrices is demonstrated with suitably chosen examples.

The paper uses analytical modification of the classical boundary integral equations (BIEs) for the Helmholtz equation to facilitate the process of practical definition of the boundary geometry. Instead of defining the boundary by means of a boundary integral, the modification makes use of Bézier curves exclusively. As a result, a new parametric integral equation system (PIES) is obtained in which boundary geometry is taken into account in original fundamental boundary solutions. Such boundary definition makes it easy to approximate boundary functions. The proposed method to obtain numerical solution of the PIES for the Helmholtz equation is characterized by high effectiveness.

This paper considers a kind of design of a ruled surface. The design interconnects some concepts from the fields of computer-aided geometric design (CAGD) and kinematics. Dual unit spherical Bézier-like curves on the dual unit sphere (DUS) are obtained by a novel method with respect to the control points. A dual unit spherical Bézier-like curve corresponds to a ruled surface by using Study’s transference principle and closed ruled surfaces are determined via control points and also, integral invariants of these surfaces are investigated. Finally, the results are illustrated by several examples and the motion interpolation was shown as an embodiment of this method.

Classical Bézier curves (the parabolic case) with the natural barycentric description are considered. The explicit expressions for lenght, curvature, shape etc. are given. It is introduced the notion trajectory of Bézier curve and by means of the projective isotomic conjugation the corresponding envelope conic surfaces in the space are obtained.

The main goal of this article is to give an almost algorithmic way to construct almost complex structures over quaternionic Hermitian vector spaces by means of the construction of a Bézier curve and its related objects.