Rotation Transforms for Computer Graphics

pp 67-84


  • John VinceAffiliated withBournemouth University

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To ensure that this book reflects modern developments in mathematics, this chapter introduces the reader to the concepts of geometric algebra – in particular, the rotational qualities of bivectors. The chapter begins with the trigonometric and vector basis for Grassmann’s algebra, that include the inner and outer products that become united by Clifford’s geometric product. These are explored in 2D and 3D. The chapter then covers the axioms associated with geometric algebra, notation and the new topics of grades, pseudoscalars and multivectors. Traditional vector analysis does not support division by a vector, however, geometric algebra does, and is described with practical examples. The imaginary properties of the outer product are explained, which lead to the rotational properties of bivectors. For completeness, the chapter covers duality, the relationship between the vector product and the outer product, and the relationship between quaternions and bivectors. The chapter concludes with a summary and a list of the multivector operations covered.