Special Issues of the Journal of Symbolic Computation

Parametric Algebraic Curves and Applications

Report by Franz Winkler, RISC, Linz (Austria)

Guest editors:

A special issue of the JSC on "Parametric Algebraic Curves and Applications" will appear soon, i.e. either in the last issue of 1996 or one of the first issues of 1997.
The main topics treated in this issue are implicitization and parametrization problems and applications in computer aided geometric design. According to this classification, three different groups of papers can be distinguished.
The first group is concerned with implicitization of parametrically presented curves and surfaces.
Starting from a parametrization of an algebraic curve or surface we might be interested in deriving algebraic equations implicitly defining the given geometric object. This is the problem of implicitization, which amounts to a special problem in polynomial elimination theory. Gonzalez-Vega, instead of using elimination methods such as resultants or Gröbner bases, proposes to use certain symmetric functions, Newton sums, to solve these special elimination problems.
Sederberg, Goldman, and Du propose the use of moving algebraic curves for implicitizing algebraic curves. In the absence of base points, moving lines turn out to be sufficient for implicitizing rational curves.
Circular curves are traced by a point on a circle rotating around a center, which again is a point on a circle rotating aroung a center, which again ... . Such circular curves can be given parametrically in terms of sines and cosines. Hong presents an implicitization algorithm for such parametrically given curves.
The second group of papers is concerned with the inverse problem, namely parametrization of implicitly given algebraic curves and surfaces, as well as reparametrization.
Sendra and Winkler continue their work on deriving a parametrization algorithm for algebraic curves. Whenever an algebraic curve can be parametrized, then there is a variety of different parametrizations. So one wants to look for a parametrization with good properties, such as being defined with only rational or real coefficients. A theorem by Hilbert and Hurwitz is generalized, which allows to reduce such optimality questions to investigations of conics.
Alternatively, van Hoeij presents an algebraic approach based on canonical divisors. From a basis of the linear space of a rational divisor on a curve , one gets a bijective morphism from to a conic. So questions such as finding points or parametrizations over low degree algebraic extensions can be investigated for the conic and transferred to the curve .
Adjoint curves play an important rôle in the geometry of algebraic curves, in particular in parametrization. They are curves passing with certain multiplicity through the singularities of an algebraic curve. From this geometric specification one can derive linear conditions on the coefficients of the adjoints. The problem is that also neighboring singularities have to be considered, which means application of quadratic transforms blowing up the degree. Mnuk analyzes the problem of determining adjoints and describes a purely algebraic approach.
Real varities are the most common ones in geometric design. However, since algebraic geometry is mainly developed over algebraically closed fields, often reality questions are not treated. Recio and Sendra investigate the problem of deciding whether a parametrized complex curve is real, and if so how to reparametrize it over the reals.
Finally, in this group of papers we have two contributions that analyze the rationality and provide parametrization algorithms for special algebraic varieties such as canal surfaces or generalized offsets.
Consider a space curve parametrically given by , and a radius function . A canal surface is the envelope of the spheres with center at and radius . In this situation Peternell and Pottmann show that canal surfaces are rationally parametrizable, and a parametrization algorithm from them is presented.
The generalized offset to a hypersurface at distance is essentially the Zariski closure of the set of intersection points of the spheres with center on and radius , and the lines obtained when applying a direct isometry to the normal lines to at the center of the spheres. Arrondo, Sendra and Sendra characterize the unirationality of the components of the generalized offsets to hypersurfaces, and provide algorithms for computing rational parametrizations when these components are parametric.
The last group of papers treats more applied problems in the area of computer aided geometric design, such as constraint solvers or numerical approaches.
The goal in the paper by Hoffmann and Joan-Arinyo is to enhance constructive geometric constrain solvers with the capability of managing functional relationships between dimension variables. A reduction on clusters of geometric elements is introduced so that geometric constraints can be propagated between such clusters, and ultimately satisfied. Termination of this reduction system is established, and if the problem is not geometrically overconstrained then reduction leads to unique normal forms.
Bézier curves are parametric curves widely used in computer aided geometric design. Farouki investigates conic approximations to offsets of conic Bézier curves. Exact upper bounds on the error incurred in such approximations are derived.
Symbolic and numerical techniques are combined by Bajaj and Xu to construct spline approximations of real algebraic surfaces. This is achieved by a novel triangular expansion scheme on the surface conforming to point and curve singularities on the surface.

Special issues of the JSC

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