Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants

We revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory. Based on the computation of a superset of the implicit support, implicitization is reduced to computing the nullspace of a numeric matrix. The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points. Our support prediction is based on sparse (or toric) resultant theory, in order to exploit the sparsity of the input and the output. Our method may yield a multiple of the implicit equation: we characterize and quantify this situation by relating the nullspace dimension to the predicted support and its geometry. In this case, we obtain more than one multiples of the implicit equation; the latter can be obtained via multivariate polynomial gcd (or factoring). All of the above techniques extend to the case of approximate computation, thus yielding a method of sparse approximate implicitization, which is important in tackling larger problems. We discuss our publicly available Maple implementation through several examples, including the benchmark of bicubic surface. For a novel application, we focus on computing the discriminant of a multivariate polynomial, which characterizes the existence of multiple roots and generalizes the resultant of a polynomial system. This yields an efficient, output-sensitive algorithm for computing the discriminant polynomial

Implicitization of curves and (hyper)surfaces using predicted support

We reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation. For predicting the implicit support, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory. Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial. We implement our methods on Maple, and some on Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces. We apply our approach to approximate implicitization, and quantify the accuracy of the approximate output, which turns out to be satisfactory on all tested examples; we also relate our measures to Hausdorff distance. In building a square or rectangular matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complexes offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice. We compare our prototype to existing software and find that it is rather competitive

Sparse implicitization by interpolation: Geometric computations using matrix representations

Based on the computation of a superset of the implicit support, implicitization of a parametrically given hyper-surface is reduced to computing the nullspace of a numeric matrix. Our approach exploits the sparseness of the given parametric equations and of the implicit polynomial. In this work, we study how this interpolation matrix can be used to reduce some key geometric predicates on the hyper-surface to simple numerical operations on the matrix, namely membership and sidedness for given query points. We illustrate our results with examples based on our Maple implementation

Reverse engineering of CAD models via clustering and approximate implicitization

In applications like computer aided design, geometric models are often represented numerically as polynomial splines or NURBS, even when they originate from primitive geometry. For purposes such as redesign and isogeometric analysis, it is of interest to extract information about the underlying geometry through reverse engineering. In this work we develop a novel method to determine these primitive shapes by combining clustering analysis with approximate implicitization. The proposed method is automatic and can recover algebraic hypersurfaces of any degree in any dimension. In exact arithmetic, the algorithm returns exact results. All the required parameters, such as the implicit degree of the patches and the number of clusters of the model, are inferred using numerical approaches in order to obtain an algorithm that requires as little manual input as possible. The effectiveness, efficiency and robustness of the method are shown both in a theoretical analysis and in numerical examples implemented in Python

A univariate resultant based implicitation algorithm for surfaces

In this paper, we present a new algorithm for computing the implicit equation of a rational surface V from a rational parametrization P(t). The algorithm is valid independent of the existence of base points, and is based on the computation of polynomial gcds and univariate resultants. Moreover, we prove that the resultant-based formula provides a power of the implicit equation. In addition, performing a suitable linear change of parameters, we prove that this power is indeed the degree of the rational map induced by the parametrization. We also present formulas for computing the partial degrees of the implicit equation.Ministerio de Educación y cienci

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

Implicitization, Interpolation, and Syzygies

Η αλγεβρικοποίηση καμπυλών και επιφανειών είναι μία θεμελιώδης μετατροπή στην αναπαράσταση γεωμετρικών αντικειμένων από παραμετρική μορφή ή αναπαράσταση νέφους σημείων σε μία αλγεβρική αναπαράσταση, και ειδικότερα ως το μηδενοσύνολο ενός (ή περισσότερων) πολυωνυμικών εξισώσεων. Αυτή η διπλωματική εργασία ερευνά τρία ερωτήματα σχετικά με την έκφραση αυτής της αλγεβρικής αναπαράστασης καμπύλης ή επιφάνειας. Αρχικά, θεωρούμε τη μέθοδο της αραιής παρεμβολής για την αλγεβρικοποίηση: Όταν η βάση του πυρήνα του πίνακα παρεμβολής είναι σε ανοιγμένη κλιμακωτή μορφή, η αναλυτική εξίσωση μπορεί να ληφθεί άμεσα, χωρίς να απαιτούμε υπολογισμούς όπως ΜΚΔ πολυωνύμων πολλών μεταβλητών ή παραγοντοποίηση. Ως δεύτερη συνεισφορά, εξετάζουμε και αξιολογούμε μία αριθμητική μέθοδο που υπολογίζει ένα πολλαπλάσιο της αναλυτικής εξίσωσης, η οποία βασίζεται στη μέθοδο των δυνάμεων. Η τρίτη συνεισφορά αυτής της διπλωματικής εργασίας είναι να προσφέρουμε μία μέθοδο για τον υπολογισμό μίας αναπαράστασης μητρώου μίας ρητής δισδιάστατης ή τρισδιάστατης καμπύλης, ή μίας τρισδιάστατης επιφάνειας, όταν μας δίνεται μόνο ένα επαρκές σύνολο σημείων (νέφος σημείων) πάνω στο αντικείμενο με τέτοιον τρόπο ώστε η τιμή της παραμέτρου να είναι γνωστή ανά σημείο. Η μέθοδός μας επεκτείνει την προσέγγιση των αλγεβρικών συζυγιών για το πρόβλημα της αλγεβρικοποίησης επιφανειών και καμπυλών στην περίπτωση που η παραμετροποίηση δεν δίνεται αλλά υποτίθεται.Implicitization is a fundamental change of representation of geometric objects from a parametric or point cloud representation to an implicit form, namely as the zero set of one (or more) polynomial equation. This thesis examines three questions related to expressing the implicit equation of a curve or a surface. First, we consider a sparse interpolation method for implicitization: When the basis of the kernel of the interpolation matrix is in reduced row echelon form, the implicit equation can be readily obtained, without demanding computations such as multivariate polynomial GCD or factoring. As a second contribution, a numeric method that computes a multiple of the implicit equation based on the power method is tested and evaluated. The third contribution of this thesis is to provide a method for computing a matrix representation of a rational planar or space curve, or a rational surface, when we are only given a sufficiently large sample of points (point cloud) on the object in such a way that the value of the parameter is known per point. Our method extends the approach of algebraic syzygies for implicitization to the case where the parameterization is not given but only assumed