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Geometric Algebra [M8]

This course introduces the mathematical language of geometric algebra as a tool for analysing problems in physics and geometry.

This course introduces the mathematical language of geometric algebra as a tool for analysing problems in physics and geometry. Students will learn the basic rules of the algebra, which follow from the fundamental ability to multiply together two vectors in an invertible way. Geometric algebra is a rapidly evolving subject, and the later lectures will cover advances made in the last decade. We will also cover how to code up geometric algebra algorithms, and how to use the language to reason about geometric problems at a higher level.

The course will cover the following topics:

  • Geometric algebra in 2 dimensions and the link with complex numbers

  • Geometric algebra in 3 dimensions, the role of the cross product and duality and the origin of the quaternion algebra.

  • Efficient ways to handle rotations in 3 dimensions – why game physics engines always employ quaternions.

  • Projective geometry and geometric algebra in 4 dimensions.

  • The algebraic structure of geometric algebra in arbitrary dimensions.

  • Schemes for coding up geometric algebra manipulations.

  • The geometric algebra of space-time – the space-time algebra.

  • Geometric calculus – Cauchy, Dirac and Maxwell unified in a single operator.

  • Conformal geometry and a new way to understand Euclidean geometry.

  • Translations, rotations and dilations handled in a unified framework.