Conics, Their Pencils and Intersections in Geometric Algebra

IF 1.1 2区数学 Q2 MATHEMATICS, APPLIED

Advances in Applied Clifford Algebras Pub Date : 2024-11-05 DOI:10.1007/s00006-024-01356-5

Clément Chomicki, Stéphane Breuils, Venceslas Biri, Vincent Nozick

引用次数: 0

Abstract

This paper presents an approach for extracting points from conic intersections by using the concept of pencils. This method is based on QC2GA—the two-dimensional version of QCGA (Quadric Conformal Geometric Algebra)—that is demonstrated to be equivalent to GAC (Geometric Algebra for Conics). A new interpretation of QC2GA and its objects based on pencils of conics and point space elements is presented, enabling the creation, constraining, and exploitation of pencils of conics. A Geometric Algebra method for computing the discriminants and center point of a conic will also be presented, enabling the proposition of an algorithm for extracting points from a conic intersection object.

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几何代数中的圆锥曲线、其铅笔和交点

本文提出了一种利用铅笔概念从圆锥交点提取点的方法。该方法基于 QC2GA--QCGA（Quadric Conformal Geometric Algebra，四元共形几何代数）的二维版本--经证明等同于 GAC（Geometric Algebra for Conics，圆锥几何代数）。基于圆锥曲线铅笔和点空间元素，提出了对 QC2GA 及其对象的新解释，从而能够创建、约束和利用圆锥曲线铅笔。还将介绍计算圆锥的判别式和中心点的几何代数方法，从而提出从圆锥交点对象中提取点的算法。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Advances in Applied Clifford Algebras 数学-物理：数学物理

CiteScore

2.20

自引率

13.30%

发文量

审稿时长

3 months

期刊介绍： Advances in Applied Clifford Algebras (AACA) publishes high-quality peer-reviewed research papers as well as expository and survey articles in the area of Clifford algebras and their applications to other branches of mathematics, physics, engineering, and related fields. The journal ensures rapid publication and is organized in six sections: Analysis, Differential Geometry and Dirac Operators, Mathematical Structures, Theoretical and Mathematical Physics, Applications, and Book Reviews.