Symmetry Detection of Rational Space Curves from their Curvature and Torsion

We present a novel, deterministic, and efficient method to detect whether a given rational space curve is symmetric. By using well-known differential invariants of space curves, namely the curvature and torsion, the method is significantly faster, simpler, and more general than an earlier method addressing a similar problem. To support this claim, we present an analysis of the arithmetic complexity of the algorithm and timings from an implementation in Sage.Comment: 25 page

The implicit equation of a canal surface

A canal surface is an envelope of a one parameter family of spheres. In this paper we present an efficient algorithm for computing the implicit equation of a canal surface generated by a rational family of spheres. By using Laguerre and Lie geometries, we relate the equation of the canal surface to the equation of a dual variety of a certain curve in 5-dimensional projective space. We define the \mu-basis for arbitrary dimension and give a simple algorithm for its computation. This is then applied to the dual variety, which allows us to deduce the implicit equations of the the dual variety, the canal surface and any offset to the canal surface.Comment: 26 pages, to be published in Journal of Symbolic Computatio

Mini-Workshop: Surface Modeling and Syzygies

The problem of determining the implicit equation of the image of a rational map φ : P2 99K P3 is of theoretical interest in algebraic geometry, and of practical importance in geometric modeling. There are essentially three methods which can be applied to the problem: Gröbner bases, resultants, and syzygies. Elimination via Gröbner basis methods tends to be computationally intensive and, being a general tool, is not adapted to the geometry of specific problems. Thus, it is primarily the latter two techniques which are used in practice. This is an extremely active area of research where many different perspectives come into play. The mini-workshop brought together a diverse group of researchers with different areas of expertise

Affine equivalences of surfaces of translation and minimal surfaces, and applications to symmetry detection and design

We introduce a characterization for affine equivalence of two surfaces of translation defined by either rational or meromorphic generators. In turn, this induces a similar characterization for minimal surfaces. In the rational case, our results provide algorithms for detecting affine equivalence of these surfaces, and therefore, in particular, the symmetries of a surface of translation or a minimal surface of the considered types. Additionally, we apply our results to designing surfaces of translation and minimal surfaces with symmetries, and to computing the symmetries of the higher-order Enneper surfaces.publishedVersio

Offsets, Conchoids and Pedal Surfaces

We discuss three geometric constructions and their relations, namely the offset, the conchoid and the pedal construction. The offset surface F d of a given surface F is the set of points at fixed normal distance d of F. The conchoid surface G d of a given surface G is obtained by increasing the radius function by d with respect to a given reference point O. There is a nice relation between offsets and conchoids: The pedal surfaces of a family of offset surfaces are a family of conchoid surfaces. Since this relation is birational, a family of rational offset surfaces corresponds to a family of rational conchoid surfaces and vice versa. We present theoretical principles of this mapping and apply it to ruled surfaces and quadrics. Since these surfaces have rational offsets and conchoids, their pedal and inverse pedal surfaces are new classes of rational conchoid surfaces and rational offset surfaces

The Relation Between Offset and Conchoid Constructions

The one-sided offset surface Fd of a given surface F is, roughly speaking, obtained by shifting the tangent planes of F in direction of its oriented normal vector. The conchoid surface Gd of a given surface G is roughly speaking obtained by increasing the distance of G to a fixed reference point O by d. Whereas the offset operation is well known and implemented in most CAD-software systems, the conchoid operation is less known, although already mentioned by the ancient Greeks, and recently studied by some authors. These two operations are algebraic and create new objects from given input objects. There is a surprisingly simple relation between the offset and the conchoid operation. As derived there exists a rational bijective quadratic map which transforms a given surface F and its offset surfaces Fd to a surface G and its conchoidal surface Gd, and vice versa. Geometric properties of this map are studied and illustrated at hand of some complete examples. Furthermore rational universal parameterizations for offsets and conchoid surfaces are provided

Computing the topology of a planar or space hyperelliptic curve

We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose topology is easy to compute, under a birational mapping of the plane or the space. We report on a {\tt Maple} implementation of these algorithms, and present several examples. Complexity and certification issues are also discussed.Comment: 34 pages, lot of figure

Darboux cyclides and webs from circles

Motivated by potential applications in architecture, we study Darboux cyclides. These algebraic surfaces of order a most 4 are a superset of Dupin cyclides and quadrics, and they carry up to six real families of circles. Revisiting the classical approach to these surfaces based on the spherical model of 3D Moebius geometry, we provide computational tools for the identification of circle families on a given cyclide and for the direct design of those. In particular, we show that certain triples of circle families may be arranged as so-called hexagonal webs, and we provide a complete classification of all possible hexagonal webs of circles on Darboux cyclides.Comment: 34 pages, 20 figure

On the computation of singularities of parametrized ruled surfaces

Given a ruled surface V defined in the standard parametric form P(t1, t2), we present an algorithm that determines the singularities (and their multiplicities) of V from the parametrization P. More precisely, from P we construct an auxiliary parametric curve and we show how the problem can be simplified to determine the singularities of this auxiliary curve. Only one univariate resultant has to be computed and no elimination theory techniques are necessary. These results improve some previous algorithms for detecting singularities for the special case of parametric ruled surfaces.Ministerio de Ciencia, Innovación y Universidade