Geometric Algebra
Title: Geometric Algebra
Authors: Chris Doran
Abstract: Geometric algebra is a mathematical framework that unifies the study of points, lines, circles, planes, and spheres. It treats these elements and the transformations acting on them in a unified algebraic framework. This article explores the concept of conformal geometric algebra, a powerful technique for creating smooth, "fair" interpolations between series of points. It demonstrates a set of algorithms for performing circle blends, which can achieve an arbitrary level of G-continuity, and extends these concepts to sphere blending. The advantages of this approach are illustrated, making it a valuable tool for computer graphics applications.
Keywords: spline, geometry, geometric algebra, conformal
Main Research Question: How can we create smooth, "fair" interpolations between series of points using geometric algebra?
Methodology: The study uses conformal geometric algebra, a 5-dimensional representation of points in space. This framework treats straight lines and circles in a single, unified manner, simplifying the process of interpolation. The article presents algorithms for circle blending and sphere blending, demonstrating their effectiveness in creating smooth curves and surfaces.
Results: The research shows that conformal geometric algebra can be used to create smooth, "fair" interpolations between series of points. The algorithms presented can achieve an arbitrary level of G-continuity, making them suitable for a range of applications, including camera trajectories and surface interpolations.
Implications: The use of conformal geometric algebra for circle and sphere blending offers a powerful tool for computer graphics applications. It simplifies the process of creating smooth curves and surfaces, making it easier to design aesthetically pleasing shapes and structures. The techniques presented in this article can be applied in various fields, including architecture, automotive design, and 3D animation.
Link to Article: https://arxiv.org/abs/0310017v1 Authors: arXiv ID: 0310017v1