Related papers: Symbolic and User-friendly Geometric Algebra Routi…
Designing software systems for Geometric Computing applications can be a challenging task. Software engineers typically use software abstractions to hide and manage the high complexity of such systems. Without the presence of a unifying…
This position paper delves into the transformative role of Geometric Algebra (GA) in advancing specific areas of Computer Graphics (CG) and Extended Reality (XR), particularly in character animation, rendering, rigging, neural rendering,…
Geometric Algebra (GA) presents challenges to learners due to its highly abstract mathematical structure and complex operational rules, as translating algebraic manipulations into concrete geometric interpretations is a non-intuitive…
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…
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…
The availability of low cost sensors has led to an unprecedented growth in the volume of spatial data. However, the time required to evaluate even simple spatial queries over large data sets greatly hampers our ability to interactively…
Distributional and neural approaches to natural language semantics have been built almost exclusively on conventional linear algebra: vectors, matrices, tensors, and the operations that accompany them. These methods have achieved remarkable…
Problems involving geometric data arise in physics, chemistry, robotics, computer vision, and many other fields. Such data can take numerous forms, for instance points, direction vectors, translations, or rotations, but to date there is no…
We introduce Graphical Algebraic Geometry (GAG), a family of diagrammatic languages extending the Graphical Linear Algebra programme. We construct several languages within this family and prove that they are universal and complete for the…
This paper introduces GA-Unity, the first Unity package specifically designed for seamless integration of Geometric Algebra (GA) into collaborative networked applications. Indeed, in such contexts, it has been demonstrated that using…
Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
A tutorial of the Mathematica package CGAlgebra, for conformal geometric algebra calculations is presented. Using rule-based programming, the 5-dimensional conformal geometric algebra is implemented and defined functions simplify the…
Many generative models attempt to replicate the density of their input data. However, this approach is often undesirable, since data density is highly affected by sampling biases, noise, and artifacts. We propose a method called SUGAR…
The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gr\"{o}bner) methods are severely limited by algorithmic issues…
Categories and categorical structures are increasingly recognized as useful abstractions for modeling in science and engineering. To uniformly implement category-theoretic mathematical models in software, we introduce GATlab, a…
Alternative mathematical explorations in quantum computing can be of great scientific interest, especially if they come with penetrating physical insights. In this paper, we present a critical revisitation of our geometric (Clifford)…
The Geometric Algebra Transformer (GATr) is a versatile architecture for geometric deep learning based on projective geometric algebra. We generalize this architecture into a blueprint that allows one to construct a scalable transformer…
What symbolic format should an LLM emit for reliable 3D scene editing from natural language, and does algebraic structure help beyond compact syntax? We evaluate Conformal Geometric Algebra (CGA) as a compact symbolic interface against a…
The art of quantum algorithm design is highly nontrivial. Grover's search algorithm constitutes a masterpiece of quantum computational software. In this article, we use methods of geometric algebra (GA) and information geometry (IG) to…
Conformal Geometric Algebra (CGA) is a framework that allows the representation of objects, such as points, planes and spheres, and deformations, such as translations, rotations and dilations as uniform vectors, called multivectors. In this…