arXiv subjects

Charles G. Gunn

Publications and source records attributed to Charles G. Gunn.

5 recordsLinked to original sources

arXiv2022

A bit better: Variants of duality in geometric algebras with degenerate metrics

Multiplication by the pseudoscalar $\mathbf{I}$ has been traditionally used in geometric algebra to perform non-metric operations such as calculating coordinates and the regressive product. In algebras with degenerate metrics, such as euclidean PGA $P(\mathbb{R}^*_{3,0,1})$, this approach breaks down, leading to a search for non-metric forms of duality. The article compares the dual coordinate map $J: G \rightarrow G^*$, a double algebra duality, and Hodge duality $H: G \rightarrow G $, a single algebra duality for this purpose. While the two maps are computationally identical, only $J$ is coordinate-free and provides direct support for geometric duality, whereby every geometric primitive appears twice, once as a point-based and once as a plane-based form, an essential feature not only of projective geometry but also of euclidean kinematics and dynamics. Our analysis concludes with a proposed duality-neutral software implementation, requiring a single bit field per multi-vector.

math.GM↗

arXiv2020

Projective geometric algebra: A new framework for doing euclidean geometry

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 operations; compact, polymorphic syntax for euclidean formulas and constructions; a single intuitive sandwich form for isometries; native support for automatic differentiation; and tight integration of kinematics and rigid body mechanics. Inclusion of vector, quaternion, dual quaternion, and exterior algebras as sub-algebras simplifies the learning curve and transition path for experienced practitioners. On the practical side, it can be efficiently implemented, while its rich syntax enhances programming productivity. The basic ideas are introduced in the 2D context; the 3D treatment focus on selected topics. Advantages to traditional approaches are collected in a table at the end. The article aims to be a self-contained introduction for practitioners of euclidean geometry and includes numerous examples, figures, and tables.

math.GM↗

arXiv2020

Course notes Geometric Algebra for Computer Graphics, SIGGRAPH 2019

What is the best representation for doing euclidean geometry on computers? These notes from a SIGGRAPH 2019 short course entitled "Geometric algebra for computer graphics" introduce projective geometric algebra (PGA) as a modern framework for this task. PGA features: uniform representation of points, lines, and planes; robust, parallel-safe join and meet operations; compact, polymorphic syntax for euclidean formulas and constructions; a single intuitive sandwich form for isometries; native support for automatic differentiation; and tight integration of kinematics and rigid body mechanics. PGA includes vector, quaternion, dual quaternion, and exterior algebras as sub-algebras, simplifying the learning curve and transition path for experienced practitioners. On the practical side, it can be efficiently implemented, while its rich syntax enhances programming productivity. The basic ideas are introduced in the 2D context and developed selectively for 3D. Advantages to traditional approaches are collected in a table at the end. The article aims to be a self-contained introduction for practitioners of euclidean geometry and includes numerous examples, formulas, figures, and tables.

cs.GR↗

arXiv2016

Doing euclidean plane geometry using projective geometric algebra

The article presents a new approach to euclidean plane geometry based on projective geometric algebra (PGA). It is designed for anyone with an interest in plane geometry, or who wishes to familiarize themselves with PGA. After a brief review of PGA, the article focuses on $\mathbf{P}(\mathbb{R}^*_{2,0,1})$, the PGA for euclidean plane geometry. It first explores the geometric product involving pairs and triples of basic elements (points and lines), establishing a wealth of fundamental metric and non-metric properties. It then applies the algebra to a variety of familiar topics in plane euclidean geometry and shows that it compares favorably with other approaches in regard to completeness, compactness, practicality, and elegance. The seamless integration of euclidean and ideal (aka infinite) elements forms an essential and novel feature of the treatment. Numerous figures accompany the text. For readers with the requisite mathematical background, a self-contained coordinate-free introduction to the algebra is provided in an appendix.

math.GM↗

arXiv2016

Geometric Algebras for Euclidean Geometry

The discussion of how to apply geometric algebra to euclidean $n$-space has been clouded by a number of conceptual misunderstandings which we first identify and resolve, based on a thorough review of crucial but largely forgotten themes from $19^{th}$ century mathematics. We then introduce the dual projectivized Clifford algebra $\mathbf{P}(\mathbb{R}^*_{n,0,1})$ (euclidean PGA) as the most promising homogeneous (1-up) candidate for euclidean geometry. We compare euclidean PGA and the popular 2-up model CGA (conformal geometric algebra), restricting attention to flat geometric primitives, and show that on this domain they exhibit the same formal feature set. We thereby establish that euclidean PGA is the smallest structure-preserving euclidean GA. We compare the two algebras in more detail, with respect to a number of practical criteria, including implementation of kinematics and rigid body mechanics. We then extend the comparison to include euclidean sphere primitives. We conclude that euclidean PGA provides a natural transition, both scientifically and pedagogically, between vector space models and the more complex and powerful CGA.

math.GM↗