Miroslav Lávička

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.

On rational Minkowski Pythagorean hodograph curves

Minkowski Pythagorean hodograph curves are polynomial curves with polynomial speed, measured with respect to Minkowski norm. Curves of this special class are particularly well suited for representing medial axis transforms of planar domains. In the present paper we generalize this polynomial class to a rational class of curves in Minkowski 3-space. We show that any rational Minkowski Pythagorean hodograph curve can be obtained in terms of its associated planar rational Pythagorean hodograph curve and an additional rational function. Moreover, both in the original polynomial and new rational case, we investigate the close relationship between these associated curves in Euclidean plane and Minkowski space.

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.

A symbolic-numerical approach to approximate parameterizations of space curves using graphs of critical points

A simple algorithm for computing an approximate parameterization of real space algebraic curves using their graphs of critical points is designed and studied in this paper. The first step is determining a suitable space graph which contains all critical points of a real algebraic space curve C implicitly defined as the complete intersection of two surfaces. The construction of this graph is based on one projection of C in a general position onto an xy-plane and on an intentional choice of vertices. The second part of the designed method is a computation of a spline curve which replaces the edges of the constructed graph by segments of a chosen free-form curve. This step is formulated as an optimization problem when the objective function approximates the integral of the squared Euclidean distance of the constructed approximate curve to the intersection curve. The presented method, based on combining symbolic and numerical steps to the approximation problem, provides approximate parameterizations of space algebraic curves from a small number of approximating arcs. It may serve as a first step to several problems originating in technical practice where approximation curve parameterizations are needed.

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.

Determining surfaces of revolution from their implicit equations

Results of number of geometric operations are in many cases surfaces described implicitly. Then it is a challenging task to recognize the type of the obtained surface, find its characteristics and for the rational surfaces compute also their parameterizations. In?this contribution we will focus on surfaces of revolution. These objects, widely used in geometric modelling, are generated by rotating a generatrix around a given axis. If the generatrix is an algebraic curve then so is also the resulting surface, described uniquely by a polynomial which can be found by some well-established implicitation technique. However, starting from a polynomial it is not known how to decide if the corresponding algebraic surface is rotational or not. Motivated by this, our goal is to formulate a simple and efficient algorithm whose input is a?polynomial with the coefficients from some subfield of R and the output is the answer whether the shape is a surface of revolution. In the affirmative case we also find the equations of its axis and generatrix. Furthermore, we investigate the problem of rationality and unirationality of surfaces of revolution and show that this question can be efficiently answered discussing the rationality of a certain associated planar curve.