Parametrization of translational surfaces

The algebraic translational surface is a typical modeling surface in computer aided design and architecture industry. In this paper, we give a necessary and sufficient condition for that algebraic surface having a standard parametric representation and our proof is constructive. If the given algebraic surface is translational, then we can compute a standard parametric representation for the surface

A symbolic-numeric approach for parametrizing ruled surfaces

This paper presents symbolic algorithms to determine whether a given surface (implicitly or parametrically defined) is a rational ruled surface and find a proper parametrization of the ruled surface. However, in practical applications, one has to deal with numerical objects that are given approximately, probably because they proceed from an exact data that has been perturbed under some previous measuring process or manipulation. For these numerical objects, the authors adapt the symbolic algorithms presented by means of the use of numerical techniques. The authors develop numeric algorithms that allow to determine ruled surfaces "close" to an input (not necessarily ruled) surface, and the distance between the input and the output surface is computed.Ministerio de Ciencia, Innovación y UniversidadesNational Natural Science Foundation of Chin

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

The Edgeworth-Kuiper debris disk

(Abridged) The Edgeworth-Kuiper belt with its presumed dusty debris is a natural reference for extrsolar debris disks. We employ a new algorithm to eliminate the inclination and the distance selection effects in the known TNO populations to derive expected parameters of the "true" EKB. Its estimated mass is M_EKB=0.12 M_earth, which is by a factor of \sim 15 larger than the mass of the EKB objects detected so far. About a half of the total EKB mass is in classical and resonant objects and another half is in scattered ones. Treating the debiased populations of EKB objects as dust parent bodies, we then "generate" their dust disk with our collisional code. Apart from accurate handling of collisions and direct radiation pressure, we include the Poynting-Robertson (P-R) drag, which cannot be ignored for the EKB dust disk. Outside the classical EKB, the radial profile of the optical depth approximately follows tau \sim r^-2 which is roughly intermediate between the slope predicted analytically for collision-dominated (r^-1.5) and transport-dominated (r^-2.5) disks. The cross section-dominating grain size still lies just above the blowout size (\sim 1...2 \microm), as it would without the P-R transport. However, if the EKB were by one order of magnitude less massive, the optical depth profile would fall off as tau \sim r^-3, and the cross section-dominating grain size would shift from \sim 1...2\microm to ~100 \microm. These properties are seen if dust is assumed to be generated only by known TNOs. If the solar system were observed from outside, the thermal emission flux from the EKB dust would be about two orders of magnitude lower than for solar-type stars with the brightest known infrared excesses observed from the same distance. Herschel and other new-generation facilities should reveal extrasolar debris disks nearly as tenuous as the EKB disk. The Herschel/PACS instrument should be able to detect disks at a \sim 1...2M_EKB level.Comment: 18 pages, 14 figures, accepted for publication in A&

Characterization of rational ruled surfaces

The algebraic ruled surface is a typical modeling surface in computer aided geometric design. In this paper, we present algorithms to determine whether a given implicit or parametric algebraic surface is a rational ruled surface, and in the affirmative case, to compute a standard parametric representation for the surface