104 research outputs found
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
Surfaces of osculating circles in Euclidean space
We introduce a new class of surfaces in Euclidean 3-space, called surfaces of osculating circles, using the concept of osculating circle of a regular curve. These surfaces contain a uniparametric family of planar lines of curvature. In this paper, we classify those ones that are canal surfaces and Weingarten surfaces
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
CANAL HYPERSURFACES ACCORDING TO GENERALIZED BISHOP FRAMES IN 4-SPACE
In the present paper, we study the canal hypersurfaces according to generalized Bishop frames of type B (parallel transport frame), type C and type D in Euclidean 4-space and obtain the Gaussian, mean and principal curvatures of them in general form. We give some results for their flatness, minimality and we examine the Weingarten canal hypersurfaces according to these frames. Especially, we investigate the tubular hypersurfaces by taking the radius function is constant in these canal hypersurfaces
Computing the μ-bases of algebraic monoid curves and surfaces
The μ-basis is a developing algebraic tool to study the expressions of rational curves and surfaces. It can play a bridge role between the parametric forms and implicit forms and show some advantages in implicitization, inversion formulas and singularity computation. However, it is difficult and there are few works to compute the μ-basis from an implicit form. In this paper, we derive the explicit forms of μ-basis for implicit monoid curves and surfaces, including the conics and quadrics which are particular cases of these entities. Additionally, we also provide the explicit form of μ-basis for monoid curves and surfaces defined by any rational parametrization (not necessarily in standard proper form). Our technique is simply based on the linear coordinate transformation and standard forms of these curves and surfaces. As a practical application in numerical situation, if an exact multiple point can not be computed, we can consider the problem of computing “approximate μ-basis” as well as the error estimation.Agencia Estatal de Investigació
Linear precision for toric surface patches
We classify the homogeneous polynomials in three variables whose toric polar
linear system defines a Cremona transformation. This classification also
includes, as a proper subset, the classification of toric surface patches from
geometric modeling which have linear precision. Besides the well-known tensor
product patches and B\'ezier triangles, we identify a family of toric patches
with trapezoidal shape, each of which has linear precision. B\'ezier triangles
and tensor product patches are special cases of trapezoidal patches
Changing representation of curves and surfaces: exact and approximate methods
Το κύριο αντικείμενο μελέτης στην παρούσα διατριβή είναι η αλλαγή αναπαράστασης
γεωμετρικών αντικειμένων από παραμετρική σε αλγεβρική (ή πεπλεγμένη) μορφή.
Υπολογίζουμε την αλγεβρική εξίσωση παρεμβάλλοντας τους άγνωστους συντελεστές
του πολυωνύμου δεδομένου ενός υπερσυνόλου των μονωνύμων του. Το τελευταίο
υπολογίζεται απο το Newton πολύτοπο της αλγεβρικής εξίσωσης που υπολογίζεται
από μια πρόσφατη μέθοδο πρόβλεψης του συνόλου στήριξης της εξίσωσης. H μέθοδος
πρόβλεψης του συνόλου στήριξης βασίζεται στην αραιή (ή τορική) απαλοιφή: το
πολύτοπο υπολογίζεται από
το Newton πολύτοπο της αραιής απαλοίφουσας αν θεωρίσουμε την παραμετροποίηση ως
πολυωνυμικό σύστημα. Στα μονώνυμα που αντιστοιχούν στα ακέραια σημεία του
Newton πολυτόπου δίνονται τιμές ώστε να σχηματίσουν έναν αριθμητικό πίνακα. Ο
πυρήνα του πίνακα αυτού, διάστασης 1 σε ιδανική περίπτωση, περιέχει τους
συντελεστές των
μονωνύμων στην αλγεβρική εξίσωση. Υπολογίζουμε τον πυρήνα του πίνακα είτε
συμβολικά είτε αριθμητικά εφαρμόζοντας την μέθοδο του singular value
decomposition (SVD). Προτείνουμε τεχνικές για να διαχειριστούμε την περίπτωση
ενός πολυδιάστατου πυρήνα το οποίο εμφανίζεται όταν το προβλεπόμενο σύνολο
στήριξης είναι ένα υπερσύνολο του
πραγματικού. Αυτό δίνει έναν αποτελεσματικό ευαίσθητο-εξόδου αλγόριθμο
υπολογισμού της αλγεβρικής εξίσωσης. Συγκρίνουμε διαφορετικές προσεγγίσεις
κατασκευής του πίνακα μέσω των λογισμικών Maple και SAGE. Στα πειράματά μας
χρησιμοποιήθηκαν ρητές
καμπύλες και επιφάνειες καθώς και NURBS. Η μέθοδός μας μπορεί να εφαρμοστεί σε
πολυώνυμα ή ρητές παραμετροποιήσεις επίπεδων καμπυλών ή (υπερ)επιφανειών
οποιασδήποτε διάστασης συμπεριλαμβανομένων και των περιπτώσεων με
παραμετροποίηση σεσημεία βάσης που εγείρουν σημαντικά ζητήματα για άλλες
μεθόδους αλγεβρικοποίησης.
Η μέθοδος έχει τον εξής περιορισμό: τα γεωμετρικά αντικείμενα πρέπει να
αναπαριστώνται από βάσεις μονωνύμων που στην περίπτωση τριγωνομετρικών
παραμετροποιήσεων θα πρέπει να μπορούν να μετασχηματιστούν σε ρητές
συναρτήσεις. Επιπλέον η τεχνική που
προτείνουμε μπορεί να εφαρμοστεί σε μη γεωμετρικά προβλήματα όπως ο
υπολογισμόςτης διακρίνουσας ενός πολυωνύμου με πολλές μεταβλητές ή της
απαλοίφουσας ενός συστήματος πολυωνύμων με πολλές μεταβλητές.The main object of study in our dissertation is the representation change of
the geometric objects
from the parametric form to implicit. We compute the implicit equation
interpolating the
unknown coefficients of the implicit polynomial given a superset of its
monomials. The latter is
derived from the Newton polytope of the implicit equation obtained by the
recently developed
method for support prediction. The support prediction method we use relies on
sparse (or
toric) elimination: the implicit polytope is obtained from the Newton polytope
of the sparse
resultant of the system in parametrization, represented as polynomials. The
monomials that
correspond to the lattice points of the Newton polytope are suitably evaluated
to build a numeric
matrix, ideally of corank 1. Its kernel contains their coefficients in the
implicit equation.
We compute kernel of the matrix either symbolically, or numerically, applying
singular value
decomposition (SVD). We propose techniques for handling the case of the
multidimensional
kernel space, caused by the predicted support being a superset of the actual.
This yields an
efficient, output-sensitive algorithm for computing the implicit equation. We
compare different
approaches for constructing the matrix in Maple and SAGE software. In our
experiments we
have used classical algebraic curves and surfaces as well as NURBS. Our method
can be
applied to polynomial or rational parametrizations of planar curves or
(hyper)surfaces of any
dimension including cases of parameterizations with base points which raise
important issues
for other implicitization methods. The method has its limits: geometric objects
have to be presented
using monomial basis; in the case of trigonometric parametrizations they have
to be
convertible to rational functions. Moreover, the proposed technique can be
applied for nongeometric
problems such as the computation of the discriminant of a multivariate
polynomial
or the resultant of a system of multivariate polynomials
- …