New Applications of Clifford’s Geometric Algebra

Abstract

The new applications of Clifford's geometric algebra surveyed in this paper include kinematics and robotics, computer graphics and animation, neural networks and pattern recognition, signal and image processing, applications of versors and orthogonal transformations, spinors and matrices, applied geometric calculus, physics, geometric algebra software and implementations, applications to discrete mathematics and topology, geometry and geographic information systems, encryption, and the representation of higher order curves and surfaces.

Basis-free Formulas for Characteristic Polynomial Coefficients in Geometric Algebras

“…The geometric algebras of vector spaces of dimensions n = 4, 5, and 6 are important for different applications in physics (the space-time algebra G 1,3 [8,13,17], the conformal space-time algebras G 4,2 and G 2,4 [7,8]), in computer science and engineering (the conformal geometric algebra G 4,1 [4,5,10,14,18]), in computer vision and computer graphics (the geometric algebra G 3,3 of projective geometry [9,16]). In particular, the characteristic polynomial coefficients are used to solve the Sylvester and Lyapunov equations in geometric algebra [22,24].…”

“…The geometric algebras of vector spaces of dimensions n = 4, 5, and 6 are important for different applications. The space-time algebra G 1,3 is widely used for applications in physics [8,13,17], the conformal geometric algebra G 4,1 is widely used in computer science and engineering [4,5,10,14,18], the geometric algebra G 3,3 of projective geometry is used in computer vision and computer graphics [9,16], the conformal space-time algebras G 4,2 and G 2,4 are used in physics [7,8].…”

On factorization of multivectors in Cl(2,1)$$ Cl\left(2,1\right) $$, by exponentials and idempotents

“…Concrete applications may therefore be to forward and reverse kinematic motions of robot arms, where such factorizations could be useful, or in drone controls. * In earlier work, the question of factorization into exponential factors, blades, and idempotents for Clifford algebras Cl(p, q), n = p + q = 1, 2 28 has been studied, as well as for Cl (3,0), Cl (1,2), and Cl(0, 3) in Hitzer. 29 This motivates us to progress by extending Hitzer 28,29 to the relatively more involved case Cl (2,1).…”

“…The importance of the polar representation of complex numbers and quaternions is widely known. Here, we endeavor to extend this approach to the higher dimensional associative Clifford geometric algebra $$$ Cl\left(2,1\right) $$$, which plays an important role in geometry, physics, and computer science 1–8 . Namely, it is the physical algrebra of 2 + 1 space‐time and the conformal geometric algebra $$$ Cl\left(1+1,1\right) $$$ of one‐dimensional Euclidean space $$$ {\mathbb{R}}^1 $$$.…”

Singularities of Serial Robots: Identification and Distance Computation Using Geometric Algebra

“…In this context, geometric algebra turns out to be very useful. In addition, it is currently applied to several problems in robot kinematics and geometry [11][12][13][14][15].…”