Related papers: Some Applications of Algebraic Curves to Computati…
Computational topology is an area that revisits topological problems from an algorithmic point of view, and develops topological tools for improved algorithms. We survey results in computational topology that are concerned with graphs drawn…
Recently, multiple formulations of vision problems as probabilistic inversions of generative models based on computer graphics have been proposed. However, applications to 3D perception from natural images have focused on low-dimensional…
A tutorial introduction to projective geometric algebra (PGA), a modern, coordinate-free framework for doing euclidean geometry. PGA features: uniform representation of points, lines, and planes; robust, parallel-safe join and meet…
We provide algorithms to reconstruct rational ruled surfaces in three-dimensional projective space from the `apparent contour' of a single projection to the projective plane. We deal with the case of tangent developables and of general…
These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra…
A quadric in $\R P^3$ cuts a curve of degree 6 on a cubic surface in $\R P^3$. The papers classifies the nonsingular curves cut in this way on non-singular cubic surfaces up to homeomorphism. Two issues new in the study related to the first…
In this survey I should like to introduce some concepts of algebraic geometry and try to demonstrate the fruitful interaction between algebraic geometry and computer algebra and, more generally, between mathematics and computer science. One…
We review some recent applications of machine learning to algebraic geometry and physics. Since problems in algebraic geometry can typically be reformulated as mappings between tensors, this makes them particularly amenable to supervised…
We contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three dimensional projective space. This problem is first turned into the computation of the finite…
Reconstructing the missing parts of a curve has been the subject of much computational research, with applications in image inpainting, object synthesis, etc. Different approaches for solving that problem are typically based on processes…
We study configurations of immersed curves in surfaces and surfaces in 3-manifolds. Among other results, we show that primitive curves have only finitely many configurations which minimize the number of double points. We give examples of…
The arithmetic of elliptic curves, namely polynomial addition and scalar multiplication, can be described in terms of global sections of line bundles on $E\times E$ and $E$, respectively, with respect to a given projective embedding of $E$…
We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let $k$ be a field and let $\mathcal{L}$ be a collection of $n$ space curves in $k^3$, with…
In a previous work of the authors, a result to algorithmically compute the topology types of the level curves of an algebraic surface, is given. From this result, here we derive applications based on level curves to determine some…
Given two general rational curves of the same degree in two projective spaces, one can ask whether there exists a third rational curve of the same degree that projects to both of them. We show that, under suitable assumptions on the degree…
This is a survey article on real algebra and geometry, and in particular on its recent applications in optimization and convexity. We first introduce basic notions and results from the classical theory. We then explain how these relate to…
Real-world applications of computational fluid dynamics often involve the evaluation of quantities of interest for several distinct geometries that define the computational domain or are embedded inside it. For example, design optimization…
In many areas of applied geometric/numeric computational mathematics, including geo-mapping, computer vision, computer graphics, finite element analysis, medical imaging, geometric design, and solid modeling, one has to compute incidences,…
Geometric algebra is an optimal frame work for calculating with vectors. The geometric algebra of a space includes elements that represent all the its subspaces (lines, planes, volumes, ...). Conformal geometric algebra expands this…
This draft summarizes some basics about geometric computer vision needed to implement efficient computer vision algorithms for applications that use measurements from at least one digital camera mounted on a moving platform with a special…