68 research outputs found
Cissoid constructions of augmented rational ruled surfaces
J.R. Sendra is member of the Research Group ASYNACS (Ref.CT-CE2019/683)Given two real affine rational surfaces we derive a criterion for deciding the rationality of their cissoid. Furthermore, when one of the surfaces is augmented ruled and the other is either an augmented ruled or an augmented Steiner surface, we prove that the cissoid is rational. Furthermore, given rational parametrizations of the surfaces, we provide a rational parametrization of the cissoid.Ministerio de EconomĂa y CompetitividadEuropean Regional Development Fun
Numerical proper reparametrization of space curves and surfaces
Simplifying rational parametrizations of surfaces is a basic problem in CAD (computer-aided design). One important way is to reduce their tracing index, called proper reparametrization. Most existing proper reparametrization work is symbolic, yet in practical environments the surfaces are usually given with perturbed coefficients hence need a numerical technique of reparametrization considering the intrinsic properness of the perturbed surfaces. We present algorithms for reparametrizing a numerically rational space curve or surface. First, we provide an efficient way to find a parametric support transformation and compute a reparametrization with proper parametric support. Second, we develop a numerical algorithm to further reduce the tracing index, where numerical techniques such as sparse interpolation and approximated GCD computations are involved. We finally provide the error bound between the given rational curve/surface and our reparametrization result.Ministerio de Ciencia, InnovaciĂłn y Universidade
Involutions of polynomially parametrized surfaces
We provide an algorithm for detecting the involutions leaving a surface
defined by a polynomial parametrization invariant. As a consequence, the
symmetry axes, symmetry planes and symmetry center of the surface, if any, can
be determined directly from the parametrization, without computing or making
use of the implicit representation. The algorithm is based on the fact, proven
in the paper, that any involution of the surface comes from an involution of
the parameter space (the real plane, in our case); therefore, by determining
the latter, the former can be found. The algorithm has been implemented in the
computer algebra system Maple 17. Evidence of its efficiency for moderate
degrees, examples and a complexity analysis are also given
Symplectomorphism groups and embeddings of balls into rational ruled 4-manifolds
Let be any rational ruled symplectic four-manifold. Given a symplectic
embedding \iota:B_{c}\into X of the standard ball of capacity into ,
consider the corresponding symplectic blow-up \tX_{\iota}. In this paper, we
study the homotopy type of the symplectomorphism group \Symp(\tX_{\iota}),
simplifying and extending the results of math.SG/0207096. This allows us to
compute the rational homotopy groups of the space \IEmb(B_{c},X) of
unparametrized symplectic embeddings of into . We also show that the
embedding space of one ball in , and the embedding space of two disjoint
balls in , if non empty, are always homotopy equivalent to the
corresponding spaces of ordered configurations. Our method relies on the theory
of pseudo-holomorphic curves in 4-manifolds, on the theory of Gromov
invariants, and on the inflation technique of Lalonde-McDuff.Comment: New title, new abstract, content now agrees with the published
version, small correction to the proof of Theorem 1.10. A sequel to the paper
SG/020709
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
Error bounded approximate reparametrization of NURBS curves
Journal ArticleThis paper reports research on solutions to the following reparametrization problem: approximate c(r(t)) by a NURBS where c is a NURBS curve and r may, or may not, be a NURBS function. There are many practical applications of this problem including establishing and exploring correspondence in geometry, creating related speed profiles along motion curves for animation, specifying speeds along tool paths, and identifying geometrically equivalent, or nearly equivalent, curve mappings. A framework for the approximation problem is described using two related algorithmic schemes. One constrains the shape of the approximation to be identical to the original curve c. The other relaxes this constraint. New algorithms for important cases of curve reparametrization are developed from within this framework. They produce results with bounded error and address approximate arc length parametrizations of curves, approximate inverses of NURBS functions, and reparametrizations that establish user specified tolerances as bounds on the Frechet distance between parametric curves
Parameterization of rational translational surfaces
A rational translational surface is a typical modeling surface used in computer-aided design and the architecture industry. In this study, we determine whether a given algebraic surface implicitly defined as V is a rational translational surface or not. This problem is reduced to finding the rational parameterizations of two space curves. More important, our discussions are constructive, and thus if V is translational, we provide a parametric representation of V of the form P(t1,t2)=P1(t1)+P2(t2).Ministerio de Ciencia, Innovacion y Universidade
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