Christos Konaxis

An output-sensitive algorithm for computing projections of resultant polytopes

We develop an incremental algorithm to compute the Newton polytope of the resultant, aka resultant polytope, or its projection along a given direction. The resultant is fundamental in algebraic elimination and in implicitization of parametric hypersurfaces. Our algorithm exactly computes vertex- and halfspace-representations of the desired polytope using an oracle producing resultant vertices in a given direction. It is output-sensitive as it uses one oracle call per vertex. We overcome the bottleneck of determinantal predicates by hashing, thus accelerating execution from 18 to 100 times. We implement our algorithm using the experimental CGAL package triangulation. A variant of the algorithm computes successively tighter inner and outer approximations: when these polytopes have, respectively, 90% and 105% of the true volume, runtime is reduced up to 25 times. Our method computes instances of 5-, 6- or 7-dimensional polytopes with 35K, 23K or 500 vertices, resp., within 2hr. Compared to tropical geometry software, ours is faster up to dimension 5 or 6, and competitive in higher dimensions.

Implicitization of curves and surfaces using predicted support

We reduce implicitization of rational parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation. For this, we may use any method for predicting the implicit support. We focus on methods that exploit input structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial. We offer a public-domain implementation of our methods, and study their numerical stability and efficiency on several classes of plane curves and surfaces, and discuss how it can be used for approximate implicitization in the setting of sparse elimination.

The maximum number of faces of the minkowski sum of three convex polytopes

We derive tight expressions for the maximum number of k-faces, 0≤k≤d-1, of the Minkowski sum, P₁+P₂+P₃, of three d-dimensional convex polytopes P₁, P₂ and P₃ in R^d, as a function of the number of vertices of the polytopes, for any d≥2. Expressing the Minkowski sum as a section of the Cayley polytope C of its summands, counting the k-faces of P₁+P₂+P₃ reduces to counting the (k+2)-faces of C which meet the vertex sets of the three polytopes. In two dimensions our expressions reduce to known results, while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of r d-polytopes in R^d, where r≥d. For d≥4, the maximum values are attained when P₁, P₂ and P₃ are d-polytopes, whose vertex sets are chosen appropriately from three distinct d-dimensional moment-like curves.

AN ORACLE-BASED, OUTPUT-SENSITIVE ALGORITHM FOR PROJECTIONS OF RESULTANT POLYTOPES

We design an algorithm to compute the Newton polytope of the resultant, known as resultant polytope, or its orthogonal projection along a given direction. The resultant is fundamental in algebraic elimination, optimization, and geometric modeling. Our algorithm exactly computes vertex- and halfspace-representations of the polytope using an oracle producing resultant vertices in a given direction, thus avoiding walking on the polytope whose dimension is alpha-n-1, where the input consists of alpha points in Z^n. Our approach is output-sensitive as it makes one oracle call per vertex and facet. It extends to any polytope whose oracle-based definition is advantageous, such as the secondary and discriminant polytopes. Our publicly available implementation uses the experimental CGAL package triangulation. Our method computes 5-, 6- and 7-dimensional polytopes with 35K, 23K and 500 vertices, respectively, within 2hrs, and the Newton polytopes of many important surface equations encountered in geometric modeling in <1sec, whereas the corresponding secondary polytopes are intractable. It is faster than tropical geometry software up to dimension 5 or 6. Hashing determinantal predicates accelerates execution up to 100 times. One variant computes inner and outer approximations with, respectively, 90% and 105% of the true volume, up to 25 times faster.

The maximum number of faces of the Minkowski sum of three convex polytopes

We derive tight expressions for the maximum number of

Minkowski Decomposition and Geometric Predicates in Sparse Implicitization

k

Geometric operations using sparse interpolation matrices

-faces,

Matrix Representations by Means of Interpolation

0\le{}k\le{}d-1

Sparse Implicitization via Interpolation

, of the Minkowski sum,

Implicitization of curves and (hyper)surfaces using predicted support

P_1+P_2+P_3