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One of the most powerful ideas in the study and classification of algebraic varieties is the notion of a model: that is, to single out an object, in the appropriate isomorphism class, with nice properties. This survey aims to define and…
In this paper we consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore…
We develop a direct and elementary (calculus-free) exposition of the famous cubic surface of revolution x^3+y^3+z^3-3xyz=1.12 pages. We have added a second elementary proof that the surface is of revolution.
In this paper we study geodesic mappings of $n$-dimensional surfaces of revolution. From the general theory of geodesic mappings of equidistant spaces we specialize to surfaces of revolution and apply the obtained formulas to the case of…
Computing occluding contours is a key building block of non-photorealistic rendering, but producing contours with consistent visibility has been notoriously challenging. This paper describes the first general-purpose smooth surface…
Embedding diagrams prove to be quite useful when learning general relativity as they offer a way of visualizing spacetime curvature through warped two dimensional (2D) surfaces. In this manuscript we present a different 2D construct that…
Translation surfaces can be defined in an elementary way via polygons, and arise naturally in in the study of various basic dynamical systems. They can also be defined as Abelian differentials on Riemann surfaces, and have moduli spaces…
This paper discusses the geometry of a surface endowed with a slope metric. We obtain necessary and sufficient conditions for any surface of revolution to admit a strongly convex slope metric. Such conditions involve certain inequalities…
Quadratic surfaces gain more and more attention among the Geometric Algebra community and some frameworks were proposed in order to represent, transform, and intersect these quadratic surfaces. As far as the authors know, none of these…
We study the rational approximation properties of special manifolds defined by a set of polynomials with rational coefficients. Mostly we will assume the case of all polynomials to depend on only one variable. In this case the manifold can…
A smooth T-surface can be thought of as a generalization of a surface of revolution in such a way that the axis of rotation is not fixed at one point but rather traces a smooth path on the base plane. Furthermore, the action, by which the…
A method is presented for computing all the affine equivalences between two rational ruled surfaces defined by rational parametrizations that works directly in parametric rational form, i.e. without computing or making use of the implicit…
We classify all surfaces with constant Gaussian curvature $K$ in Euclidean $3$-space that can be expressed as an implicit equation of type $f(x)+g(y)+h(z)=0$, where $f$, $g$ and $h$ are real functions of one variable. If $K=0$, we prove…
The simplest version of the Spin-polynomial invariants of the underlying differentiable structures of algebraic surfaces were considered and the simplest arguments were used in order to distinguish the underlying smooth structures of…
The emergence of explainability methods has enabled a better comprehension of how deep neural networks operate through concepts that are easily understood and implemented by the end user. While most explainability methods have been designed…
The linear orbit of a degree d hypersurface in $\mathbb{P}^n$ is its orbit under the natural action of PGL(n+1), in the projective space of dimension $N =\binom{n+d}{d} - 1$ parameterizing such hypersurfaces. This action restricted to a…
Polyhedral surfaces are fundamental objects in architectural geometry and industrial design. Whereas closeness of a given mesh to a smooth reference surface and its suitability for numerical simulations were already studied extensively, the…
The group PGL(3) of linear transformations of the projective plane acts naturally on the projective space parametrizing curves of a given degree. In this note we begin the study of the orbits of smooth curves under this action: we construct…
We derive parametrizations of the Delaunay constant mean curvature surfaces of revolution that follow directly from parametrizations of the conics that generate these surfaces via the corresponding roulette. This uniform treatment exploits…
Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane…