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This paper presents a method for computing interleaved additive and subtractive manufacturing operations to fabricate models of arbitrary shapes. We solve the manufacturing planning problem by searching a sequence of inverse operations that…
Given a family of rational curves depending on a real parameter, defined by its parametric equations, we provide an algorithm to compute a finite partition of the parameter space (${\Bbb R}$, in general) so that the shape of the family…
We construct an example of a real plane analytic singular metric, degenerating only at the origin, such that any gradient trajectory (respectively to this singular metric) of some well chosen function spirals around the origin. The…
In this paper, we present a novel method to draw a circle tangent to three given circles lying on a plane. Using the analytic geometry and inversion (reflection) theorems, the center and radius of the inversion circle are obtained. Inside…
In this paper, we propose a novel lower dimensional representation of a shape sequence. The proposed dimension reduction is invertible and computationally more efficient in comparison to other related works. Theoretically, the differential…
Given a two-dimensional conformal field theory with a global symmetry, we propose a method to implement an orbifold construction by taking orbits of the modular group. For the case of cyclic symmetries we find that this approach always…
A new robust algorithm for the numerical computation of biarcs, i.e. $G^1$ curves composed of two arcs of circle, is presented. Many algorithms exist but are based on geometric constructions, which must consider many geometrical…
Our goal is to present, in what we believe is the most efficient way possible, a construction of four mutually tangent circles.
We present a construction of new invariant sets for fibred polynomial dynamics with base an irrational rotation over the unit circle, called multi-curves. Furthermore, the local dynamical theory for attracting invariant curves is extended…
Surfaces and curves play an important role in geometric design. In recent years, problem of finding a surface passing through a given curve have attracted much interest. In the present paper, we propose a new method to construct a surface…
A $\textit{polygonal curve}$ is a collection of $m$ connected line segments specified as the linear interpolation of a list of points $\{p_0, p_1, \ldots, p_m\}$. These curves may be obtained by sampling points from an oriented curve in…
This study defines finite-type invariants for curves on surfaces and reveals the construction of these finite-type invariants for stable homeomorphism classes of curves on compact oriented surfaces without boundaries. These invariants are a…
Dimer models (also known as brane tilings) are special bipartite graphs on a torus $\mathbb{T}^2$. They encode the structure of the 4d $\mathcal{N} = 1$ worldvolume theories of D3 branes probing toric affine Calabi-Yau singularities.…
The square root velocity transformation provides a convenient and numerically efficient approach to functional and shape data analysis of curves. We study fundamental geometric properties of curves under this transformation. Moreover,…
We describe a method that allows, under some hypotheses, to compute all the rational points of some genus 5 curves defined over a number field. This method is used to solve some arithmetic problems that remained open.
We introduce a technique for recovering a sufficiently smooth function from its ray transform over a wide class of curves in a general region of Euclidean space. The method is based on a complexification of the underlying vector fields…
In a recent paper, the author and St\"ohr established a bound on the number of iterated Frobenius pullbacks needed to transform a non-smooth non-decomposed point on a regular geometrically integral curve into a rational point. In this note…
We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on quasiorthogonality in the Mordell-Weil lattice…
We present a novel, deterministic, and efficient method to detect whether a given rational space curve is symmetric. By using well-known differential invariants of space curves, namely the curvature and torsion, the method is significantly…
Let R be a ring. A construction method for flexible quadratic algebras with scalar involution over R is presented which unifies various classical constructions in the literature, in particular those to construct composition algebras.