Bohumír Bastl

Computing exact rational offsets of quadratic triangular Bézier surface patches

The offset surfaces to non-developable quadratic triangular Bezier patches are rational surfaces. In this paper we give a direct proof of this result and formulate an algorithm for computing the parameterization of the offsets. Based on the observation that quadratic triangular patches are capable of producing C^1 smooth surfaces, we use this algorithm to generate rational approximations to offset surfaces of general free-form surfaces.

Hermite interpolation by hypocycloids and epicycloids with rational offsets

We show that all rational hypocycloids and epicycloids are curves with Pythagorean normals and thus have rational offsets. Then, exploiting the convolution properties and (implicit) support function representation of these curves, we design an efficient algorithm for G^1 Hermite interpolation with their arcs. We show that for all regular data, there is a unique interpolating hypocycloidal or epicycloidal arc of the given canonical type.

Rational hypersurfaces with rational convolutions

The aim of this article is to focus on the investigation of such rationally parametrized hypersurfaces which admit rational convolutions generally, or in some special cases. Examples of such hypersurfaces are presented and their properties are discussed. We also aim to examine links between well-known curves and surfaces (PH/PN or LN) and objects defined and explored in this article. In addition, the paper brings a proof that the convolution surfaces of non-developable quadratic Bezier surfaces and an arbitrary rational surface are always rational.

PN surfaces and their convolutions with rational surfaces

Rationally parameterized hypersurfaces can be classified with respect to their RC properties (Rational Convolutions) with the help of the Grobner bases theory. This classification focuses on special classes of rational parameterizations which provide a rational description of convolution hypersurfaces generally (GRC parameterizations), or just in some special cases (SRC parameterizations). The main aim of this paper is to bring the theory of the so-called PN surfaces (surfaces with Pythagorean Normal vectors) and their PN parameterizations (parameterizations fulfilling the PN condition) in relation to the theory of SRC parameterizations and to show that this type of parameterizations can be further classified with respect to the degree of the construction of convolution surfaces. The connection of SRC PN parameterizations to the well-known concepts of proper and square-root parameterizations is also investigated.

Volumes with piecewise quadratic medial surface transforms: Computation of boundaries and trimmed offsets

MOS surfaces (i.e., medial surface transforms obeying a sum of squares condition) are rational surfaces in R^3^,^1 which possess rational envelopes of the associated two-parameter families of spheres. Moreover, all offsets of the envelopes admit rational parameterizations as well. Recently, it has been proved that quadratic triangular Bezier patches in R^3^,^1 are MOS surfaces. Following this result, we describe an algorithm for computing an exact rational envelope of a two-parameter family of spheres given by a quadratic patch in R^3^,^1. The paper focuses mainly on the geometric aspects of the algorithm. Since these patches are capable of producing C^1 smooth approximations of medial surface transforms of spatial domains, we use this algorithm to generate rational approximations of envelopes of general medial surface transforms. One of the main advantages of this approach to offsetting is the fact that the trimming procedure becomes considerably simpler.

G1 Hermite interpolation by PH cubics revisited

This paper deals with G^1 Hermite interpolation by the Tschirnhausen cubic. In Meek and Walton (1997a), the explicit formulas for finding an arc of Tschirnhausen cubic which interpolates given Hermite interpolation data were given. In this paper, we extend these results to more general input data and refine on the results presented in Meek and Walton (1997a). Furthermore, we present a thorough analysis of the number and the quality of the interpolants; particularly if they contain a loop or not.

Isogeometric analysis of free vibration of simple shaped elastic samples

The paper is devoted to numerical solution of free vibration problems for elastic bodies of canonical shapes by means of a spline based finite element method (FEM), called Isogeometric Analysis (IGA). It has an advantage that the geometry is described exactly and the approximation of unknown quantities is smooth due to higher-order continuous shape functions. IGA exhibits very convenient convergence rates and small frequency errors for higher frequency spectrum. In this paper, the IGA strategy is used in computation of eigen-frequencies of a block and cylinder as benchmark tests. Results are compared with the standard FEM, the Rayleigh-Ritz method, and available experimental data. The main attention is paid to the comparison of convergence rate, accuracy, and time-consumption of IGA against FEM and also to show a spline order and parameterization effects. In addition, the potential of IGA in Resonant Ultrasound Spectroscopy measurements of elastic properties of general anisotropy solids is discussed.

Blends of canal surfaces from polyhedral medial transform representations

We present a new method for constructing G1 blending surfaces between an arbitrary number of canal surfaces. The topological relation of the canal surfaces is specified via a convex polyhedron and the design technique is based on a generalization of the medial surface transform. The resulting blend surface consists of trimmed envelopes of one- and two-parameter families of spheres. Blending the medial surface transform instead of the surface itself is shown to be a powerful and elegant approach for blend surface generation. The performance of our approach is demonstrated by several examples.

Reparameterization of curves and surfaces with respect to their convolution

Given two parametric planar curves or surfaces we find their new parameterizations (which we call coherent) permitting to compute their convolution by simply adding the points with the same parameter values. Several approaches based on rational reparameterization of one or both input objects or direct computation of new parameterizations are shown. Using the Grobner basis theory we decide the simplest possible way for obtaining coherent parametrizations. We also show that coherent parameterizations exist whenever the convolution hypersurface is rational.

C1 Hermite interpolation with spatial Pythagorean-hodograph cubic biarcs

In this paper the C^1 Hermite interpolation problem by spatial Pythagorean-hodograph cubic biarcs is presented and a general algorithm to construct such interpolants is described. Each PH cubic segment interpolates C^1 data at one point and they are then joined together with a C^1 continuity at some unknown common point sharing some unknown tangent vector. Biarcs are expressed in a closed form with three shape parameters. Two of them are selected based on asymptotic approximation order, while the remaining one can be computed by minimizing the length of the biarc or by minimizing the elastic bending energy. The final interpolating spline curve is globally C^1 continuous, it can be constructed locally and it exists for arbitrary Hermite data configurations.