Takis Sakkalis

Pythagorean-hodograph space curves

We investigate the properties of polynomial space curvesr(t)={x(t), y(t), z(t)} whose hodographs (derivatives) satisfy the Pythagorean conditionx′2(t)+y′2(t)+z′2(t)≡σ2(t) for some real polynomial σ(t). The algebraic structure of thecomplete set of regular Pythagorean-hodograph curves in ℝ3 is inherently more complicated than that of the corresponding set in ℝ2. We derive a characterization for allcubic Pythagoreanhodograph space curves, in terms of constraints on the Bézier control polygon, and show that such curves correspond geometrically to a family of non-circular helices. Pythagorean-hodograph space curves of higher degree exhibit greater shape flexibility (the quintics, for example, satisfy the general first-order Hermite interpolation problem in ℝ3), but they have no “simple” all-encompassing characterization. We focus on asubset of these higher-order curves that admits a straightforward constructive representation. As distinct from polynomial space curves in general, Pythagorean-hodograph space curves have the following attractive attributes: (i) the arc length of any segment can be determined exactly without numerical quadrature; and (ii) thecanal surfaces based on such curves as spines have precise rational parameterizations.

Hermite Interpolation by Rotation-Invariant Spatial Pythagorean-Hodograph Curves

The interpolation of first-order Hermite data by spatial Pythagorean-hodograph curves that exhibit closure under arbitrary 3-dimensional rotations is addressed. The hodographs of such curves correspond to certain combinations of four polynomials, given by Dietz et al. [4], that admit compact descriptions in terms of quaternions – an instance of the “PH representation map” proposed by Choi et al. [2]. The lowest-order PH curves that interpolate arbitrary first-order spatial Hermite data are quintics. It is shown that, with PH quintics, the quaternion representation yields a reduction of the Hermite interpolation problem to three “simple” quadratic equations in three quaternion unknowns. This system admits a closed-form solution, expressing all PH quintic interpolants to given spatial Hermite data as a two-parameter family. An integral shape measure is invoked to fix these two free parameters.

Real rational curves are not “unit speed”

We prove that it is impossible to parameterize any real plane curve, other than a straight line, by rational functions of its arc length. The proof extends in a straightforward manner to real space curves of any dimension d ⩾ 3.

Analysis and applications of pipe surfaces

Abstract A pipe (or tubular) surface is the envelope of a one-parameter family of spheres with constant radii r and centers C (t) . In this paper we investigate necessary and sufficient conditions for the nonsingularity of pipe surfaces. In addition, when C (t) is a rational function, we develop an algorithmic method for the rational parametrization of such a surface. The latter is based on finding two rational functions α(t) and β(t) such that ¦ C ′ (t)¦ 2 = α 2 (t) + β 2 (t) (Lu and Pottmann, 1996).

Singular points of algebraic curves

Given an irreducible algebraic curve f(x,y)=0 of degree n>=3 with rational coefficients,we describe algorithms for determinig whether the curve is singular, and if so, isolating its singular points, computing their multiplicities, and counting the number of distinct tangents at each. The algorithms require only rational arithmetic operations on the coefficients of f(x,y)=0, and avoid the need for more abstract symbolic representations of the singular point coordinates.

The topological configuration of a real algebraic curve

This paper presents an algorithm, motivated by Morse Theory, for the topological configuration of the components of a real algebraic curve { f ( x, y ) = 0}. The running time of the algorithm is O ( n 12 ( d + log n ) 2 log n ), where n , d are the degree and maximum coefficient size of f ( x, y ).

Topological and Geometric Properties of Interval Solid Models

A solid is a connected orientable compact subset of R3 which is a 3-manifold with boundary. Moreover, its boundary consists of finitely many components, each of which is a subset of the union of finitely many almost smooth surfaces. Motivated by numerical robustness issues, we consider a finite collection of boxes, with faces parallel to the coordinate planes, which covers the boundary of the solid itself. An interval solid is the union of this collection and the solid. In this paper we develop sufficient conditions on the collection of the boxes and a 3-manifold, so that the union of the collection and the manifold is homeomorphic to the manifold itself. Finally, we outline an approach for constructing an interval solid, using interval arithmetic, homeomorphic to the solid.

SELF-INTERSECTIONS OF OFFSET CURVES AND SURFACES

In this paper we discuss the self-intersections of offset curves and surfaces and show how to determine the maximum offset distance such that the offset does not locally nor globally self-intersect including boundary effects. Examples illustrate the applicability of the analysis.

Ambient isotopic approximations for surface reconstruction and interval solids

Given a nonsingular compact 2-manifold <i>F</i> without boundary, we present methods for establishing a family of surfaces which can approximate <i>F</i> so that each approximant is ambient isotopic to <i>F</i>. The current state of the art in surface reconstruction is that both theory and practice are limited to generating a piecewise linear (PL) approximation. The methods presented here offer broader theoretical guidance for a rich class of ambient isotopic approximations. They are also used to establish sufficient conditions for an interval solid to be ambient isotopic to the solid it is approximating.The methods are based on <i>global</i> theoretical considerations and are compared to existing <i>local</i> methods. Practical implications of these methods are also presented. For the global case, a differential surface analysis is performed to find a positive number <i>ρ</i> so that the offsets <i>F_o(± ρ)</i> of <i>F</i> at distances <i>± ρ</i> are nonsingular. In doing so, a normal tubular neighborhood, <i>F(ρ)</i>, of <i>F</i> is constructed. Then, each approximant of <i>F</i> lies inside <i>F(ρ)</i>. Comparisons between these global and local constraints are given.

Isotopic approximations and interval solids

Given a nonsingular compact two-manifold F without boundary, we present methods for establishing a family of surfaces which can approximate F so that each approximant is ambient isotopic to F: The methods presented here offer broad theoretical guidance for a rich class of ambient isotopic approximations, for applications in graphics, animation and surface reconstruction. They are also used to establish sufficient conditions for an interval solid to be ambient isotopic to the solid it is approximating. Furthermore, the normals of the approximant are compared to the normals of the original surface, as these approximating normals play prominent roles in many graphics algorithms. The methods are based on global theoretical considerations and are compared to existing local methods. Practical implications of these methods are also presented. For the global case, a differential surface analysis is performed to find a positive number r so that the offsets Foð^rÞ of F at distances ^r are nonsingular. In doing so, a normal tubular neighborhood, FðrÞ; of F is constructed. Then, each approximant of F lies inside FðrÞ: Comparisons between these global and local constraints are given. q 2004 Elsevier Ltd. All rights reserved.