Search results for "Conic section"
showing 10 items of 14 documents
An annihilator-based strategy for the automatic detection of exponential polynomial spaces in subdivision
2021
Abstract Exponential polynomials are essential in subdivision for the reconstruction of specific families of curves and surfaces, such as conic sections and quadric surfaces. It is well known that if a linear subdivision scheme is able to reproduce a certain space of exponential polynomials, then it must be level-dependent, with rules depending on the frequencies (and eventual multiplicities) defining the considered space. This work discusses a general strategy that exploits annihilating operators to locally detect those frequencies directly from the given data and therefore to choose the correct subdivision rule to be applied. This is intended as a first step towards the construction of se…
Projective Architecture
2009
Sbacchi indaga su quale sia stata la reale influenza della nozione di "proiezione" nell'ambito della progettazione architettonica a partire da Guarini. Per fare ciò vengono presi in considerazione i concetti di luce e ombra così come quelli di linea astratta, piano, sezione, geometria proiettiva e prospettiva. Michele Sbacchi investigates the real influence of the notion of projection on architectural design before and during the age of Guarini. He takes into consideration concepts such as light and shadow, abstract line, plane, section, projective geometry and perspective. To do this he looks at the ideas of Gregorius Saint Vincent, Alberti, Guarini, Desargues and de l’Orme, among others.
Historical Origins of the nine-point conic -- The Contribution of Eugenio Beltrami
2020
In this paper, we examine the evolution of a specific mathematical problem, i.e. the nine-point conic, a generalisation of the nine-point circle due to Steiner. We will follow this evolution from Steiner to the Neapolitan school (Trudi and Battaglini) and finally to the contribution of Beltrami that closed this journey, at least from a mathematical point of view (scholars of elementary geometry, in fact, will continue to resume the problem from the second half of the 19th to the beginning of the 20th century). We believe that such evolution may indicate the steady development of the mathematical methods from Euclidean metric to projective, and finally, with Beltrami, with the use of quadrat…
Remarks on the Historiography of Mathematics
2021
In this paper, I examine aspects of the methodological debate that originated in 2010, when the distinguished historian of mathematics Sabetai Unguru reviewed Roshdi Rashed’s edition of the Arabic translation of Apollonius’ Conics. In his review, Unguru criticized what Rashed calls “l’usage instrumental d’une autre mathématique pour commenter une oeuvre ancienne”. I consider this debate very important and will try to place it within in the discussion of the so-called “geometric algebra” that goes back to the seventies, by tracing the contributions of the main figures who took part in it. Published Online (2021-04-30)Copyright © 2021 by Aldo Brigaglia Article PDF Link: https://jps.library.ut…
The non-degenerate Dupin cyclides in the space of spheres using Geometric Algebra
2012
International audience; Dupin cyclides are algebraic surfaces of degree 4 discovered by the French mathematician Pierre-Charles Dupin early in the 19th century and \textcolor{black}{were} introduced in CAD by R. Martin in 1982. A Dupin cyclide can be defined, in two different ways, as the envelope of a one-parameter family of oriented spheres. So, it is very interesting to model the Dupin cyclides in the space of spheres, space wherein each family of spheres can be seen as a conic curve. In this paper, we model the non-degenerate Dupin cyclides and the space of spheres using Conformal Geometric Algebra. This new approach permits us to benefit from the advantages of the use of Geometric Alge…
Strictly convex corners scatter
2017
We prove the absence of non-scattering energies for potentials in the plane having a corner of angle smaller than $\pi$. This extends the earlier result of Bl{\aa}sten, P\"aiv\"arinta and Sylvester who considered rectangular corners. In three dimensions, we prove a similar result for any potential with a circular conic corner whose opening angle is outside a countable subset of $(0,\pi)$.
An Introduction to Geometric Algebra and Conics
2016
This chapter introduces the conics and characterizes them from an algebraic perspective. While in depth geometrical aspects of the conics lie outside the scopes of this chapter, this chapter is an opportunity to revisit concepts studied in other chapters such as matrix and determinant and assign a new geometric characterization to them.
Iterative construction of Dupin cyclides characteristic circles using non-stationary Iterated Function Systems (IFS)
2012
International audience; A Dupin cyclide can be defined, in two different ways, as the envelope of an one-parameter family of oriented spheres. Each family of spheres can be seen as a conic in the space of spheres. In this paper, we propose an algorithm to compute a characteristic circle of a Dupin cyclide from a point and the tangent at this point in the space of spheres. Then, we propose iterative algorithms (in the space of spheres) to compute (in 3D space) some characteristic circles of a Dupin cyclide which blends two particular canal surfaces. As a singular point of a Dupin cyclide is a point at infinity in the space of spheres, we use the massic points defined by J.C. Fiorot. As we su…
The cyclicity of the elliptic segment loops of the reversible quadratic Hamiltonian systems under quadratic perturbations
2004
Abstract Denote by Q H and Q R the Hamiltonian class and reversible class of quadratic integrable systems. There are several topological types for systems belong to Q H ∩ Q R . One of them is the case that the corresponding system has two heteroclinic loops, sharing one saddle-connection, which is a line segment, and the other part of the loops is an ellipse. In this paper we prove that the maximal number of limit cycles, which bifurcate from the loops with respect to quadratic perturbations in a conic neighborhood of the direction transversal to reversible systems (called in reversible direction), is two. We also give the corresponding bifurcation diagram.
Affine Algebraic Varieties
2000
Algebraic geometers study zero loci of polynomials. More accurately, they study geometric objects, called algebraic varieties, that can be described locally as zero loci of polynomials. For example, every high school mathematics student has studied a bit of algebraic geometry, in learning the basic properties of conic sections such as parabolas and hyperbolas.