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

Rational patches on Darboux and isotropic cyclides and their modeling applications - Part II

Darboux and isotropic cyclides in $\mathbb{R} P^3$ are projections of intersections of certain pairs of quadrics in $\mathbb{R} P^4$. Hence they are particular cases of real Del Pezzo surfaces (of degree 4 or less), that are are known to be rational. Low-degree rational patches on these cyclides described in a form convenient for shape manipulations are the most interesting for modeling applications. Let $\mathcal{A} P^1$ be a projective line over two cases of Clifford algebras $\mathcal{A} = Cl(\mathbb{R}^3), Cl(\mathbb{R}^{2,0,1})$, generated by euclidean space $\mathbb{R}^3$ and pseudo-euclidean space $\mathbb{R}^{2,0,1}$ with signature $(++0)$. Our approach is to treat $\mathcal{A} P^1$ as an ambient space and to consider toric Bezier patches in the corresponding homogeneous coordinates. It is proved that such patches of formal degree 2 with standard and non-standard real structures cover almost all cases of real Darboux and isotropic cyclides. The corresponding implicitization and parametrization algorithms are studied. The applications are related to families of circles on surfaces (including generation of 3-webs of circles) in case of Darboux cyclides and blending of Pythagorean-normal surfaces in case of isotropic cyclides.Non UBCUnreviewedAuthor affiliation: Vilnius universityFacult

Kvaternioninės racionalios Bézier kreivės

We extended the rational Bézier model for space curve, by allowing quaternion weights. These curves are Möbius invariant and have halved degree with respect to real Bézier curves. This simplify the analysis of curves. In general, these curves are in four dimensional space. We analyze when the quadratically parameterized quaternion curve is in usual three dimensional subspace.  Darbe yra apibrėžiamos kvaternioninės racionalios Bézier kreivės. Kadangi kontroliniai taškai yra bet kokie kvaternionai, tai kreivės patenka į keturmatę erdvę. Taikymuose dažniausiai yra reikalingos kreivės trimatėje erdvėje. Darbe pagrindinis dėmesys yra sutelktas į kvartikas. Nagrinėjamas klausimas, kad jos guli trimatėje erdvėje. &nbsp

Kvaternioninės Bézier kreivės, paviršiai ir tūriai

We extended the rational Bézier construction for linear, bi-linear and threelinear map, by allowing quaternion weights. These objects are Möbius invariant and have halved degree with respect to the real parametrization. In general, these parametrizations are in four dimensional space. We analyse when a special the three-linear parametrized volume is in usual three dimensional subspace and gives three orthogonal family of Dupine cyclides.Darbe yra nagrinėjamas kvaternioninis tritiesinis Bézier tūris, kuris konstruojamas naudojant sferino kubo kontrolinius taškus. Darbe gautos formulės atitinkamų kontrolinių taškų svoriams rasti. Kadangi taikymuose dažniausiai yra reikalingi objektai trimatėje erdvėje, todėl nagrinėjamas klausimas: kada tritiesinis atvaizdis yra trimatėje erdvėje

Interpoliacijos metodas kvaternioninėms Bezier kreivėms

We study rational quaternionic-Bézier curves in three dimensional space. We construct the quadratic quaternionic-Bézier curve which interpolates five points, or three points and two tangent vectors.Darbe yra aprašytas racionalios kvaternioninės Bézier kreivės interpoliacinis uždavinys. Pagrindinisdėmesys yra sutelktas į kvaternionines konikes. Gautos sąlygos, kada jos guli trimatėje erdvėje, kasyra svarbu taikymuose

Vector separations on elliptic surfaces

The behaviour of slightly non-ideal base-gas in the magnetic field is first studied; it is shown that such a system can be considered as a model of a superconductor of the second kind. And explanation of abnormal temperature dependence of the magnetic field penetration depth in high-temperature metal-oxide superconductors is first givenAvailable from VNTIC / VNTIC - Scientific & Technical Information Centre of RussiaSIGLERURussian Federatio