Applications to 3D dynamics In this lecture we apply the geometric algebra of three-dimensional space to problems in dynamics. The first step is to replace the concept of an ‘axial vector’ with that of a bivector. Quantities like angular momentum … Continued
Geometric Algebra in 3 Dimensions In this lecture we explore the consequences of our definition of the geometric product in three dimensions. We find that Hamilton’s quaternions arise naturally in the geometric algebra of three dimensions, but that the generators … Continued
Geometric Algebra in 2 Dimensions In this lecture we introduce geometric algebra through studying various products of vectors. Our goal is to find an associative, invertible product that allows us to multiply together two vectors. Complex arithmetic helps motivate the … Continued
Lectures Lecture 1: Geometric Algebra in 2 Dimensions Lecture 2: Geometric Algebra in 3 Dimensions Lecture 3: Applications to 3D dynamics Lecture 4: Algebraic Foundations and 4D Lecture 5: Spacetime Algebra Lecture 6: Geometric Calculus Lecture 7: Implementation Lecture 8: … Continued
Unification and Implementation In this final lecture we complete the story of how conformal geometric algebra unifies many of the classical geometries, covering Euclidean, spherical, hyperbolic (non-Euclidean), projective and inversive geometries. Rotors are shown to generate rotations, translations and special … Continued
Conformal Geometric Algebra It is only fairly recently that we realised just how powerful the combination of conformal geometry and geometric algebra could be as a tool for understanding Euclidean geometry. The key ideas were introduced around 2000, and have led … Continued
Geometric Calculus In this lecture we introduce the core concept of the vector derivative. This is the natural extension of the gradient operator to geometric algebra setting, where it inherits the properties of a grade-1 vector. The separate inner and outer … Continued
Spacetime Algebra In this lecture we introduce the geometric algebra of Minkowski spacetime, or spacetime algebra. Historically spacetime algebra drove much of the early interest in geometric algebra, and this importance was recently celebrated in a new edition of David Hestenes’ … Continued
Algebraic Foundations and 4D In this lecture we set out the algebraic rules that define a geometric algebra in spaces of arbitrary dimensions. The inner and outer products are extended to their most general definition, and we look at rotors … Continued
Applications to 3D dynamics In this lecture we apply the geometric algebra of three-dimensional space to problems in dynamics. The first step is to replace the concept of an ‘axial vector’ with that of a bivector. Quantities like angular momentum … Continued