50 research outputs found
Implicitization of rational surfaces using toric varieties
A parameterized surface can be represented as a projection from a certain
toric surface. This generalizes the classical homogeneous and bihomogeneous
parameterizations. We extend to the toric case two methods for computing the
implicit equation of such a rational parameterized surface. The first approach
uses resultant matrices and gives an exact determinantal formula for the
implicit equation if the parameterization has no base points. In the case the
base points are isolated local complete intersections, we show that the
implicit equation can still be recovered by computing any non-zero maximal
minor of this matrix.
The second method is the toric extension of the method of moving surfaces,
and involves finding linear and quadratic relations (syzygies) among the input
polynomials. When there are no base points, we show that these can be put
together into a square matrix whose determinant is the implicit equation. Its
extension to the case where there are base points is also explored.Comment: 28 pages, 1 figure. Numerous major revisions. New proof of method of
moving surfaces. Paper accepted and to appear in Journal of Algebr
Recommended from our members
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
Using implicit equations of parametric curves and surfaces without computing them: Polynomial algebra by values
The availability of the implicit equation of a plane curve or of a 3D surface can be very useful in order to solve many geometric problems involving the considered curve or surface: for example, when dealing with the point position problem or answering intersection questions. On the other hand, it is well known that in most cases, even for moderate degrees, the implicit equation is either difficult to compute or, if computed, the high degree and the big size of the coefficients makes extremely difficult its use in practice.
We will show that, for several problems involving plane curves, 3D surfaces and some of their constructions (for example, offsets), it is possible to use the implicit equation (or, more precisely, its properties) without needing to explicitly determine it. We replace the computation of the implicit equation with the evaluation of the considered parameterizations in a set of points. We then translate the geometric problem in hand, into one or several generalized eigenvalue problems on matrix pencils (depending again on several evaluations of the considered parameterizations).
This is the so-called “polynomial algebra by values” approach where the huge polynomial equations coming from Elimination Theory (e.g., using resultants) are replaced by big structured and sparse numerical matrices. For these matrices there are well-known numerical techniques allowing to provide the results we need to answer the geometric questions on the considered curves and surfaces
Intersection and self-intersection of surfaces by means of Bezoutian matrices
International audienceThe computation of intersection and self-intersection loci of parameterized surfaces is an important task in Computer Aided Geometric Design. Computer algebra tools need to be developed further for computing their implicit equations. We address these problems via four resultants with separated variables. Two of them are specializations of more general ones and the others are determinantal. We give a rigorous study in these cases and provide new and useful formulas via adapted computations of Bezoutians
Determining Critical Points of Handwritten Mathematical Symbols Represented as Parametric Curves
We consider the problem of computing critical points of plane curves represented in a finite orthogonal polynomial basis. This is motivated by an approach to the recognition of hand-written mathematical symbols in which the initial data is in such an orthogonal basis and it is desired to avoid ill-conditioned basis conversions. Our main contribution is to assemble the relevant mathematical tools to perform all the necessary operations in the orthogonal polynomial basis. These include implicitization, differentiation, root finding and resultant computation
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
On the closed image of a rational map and the implicitization problem
In this paper, we investigate some topics around the closed image of a
rational map given by some homogeneous elements of the
same degree in a graded algebra . We first compute the degree of this closed
image in case is generically finite and define isolated
base points in \Proj(A). We then relate the definition ideal of to the
symmetric and the Rees algebras of the ideal , and
prove some new acyclicity criteria for the associated approximation complexes.
Finally, we use these results to obtain the implicit equation of in case
is a hypersurface, \Proj(A)=\PP^{n-2}_k with a field, and base points
are either absent or local complete intersection isolated points.Comment: 43 pages, revised version. To appear in Journal of Algebr