Logarithm of multivector in real 3D Clifford algebras

Abstract

Closed form expressions for a logarithm of general multivector (MV) in basis-free form in real geometric algebras (GAs) Clp,q are presented for all n = p + q = 3. In contrast to logarithm of complex numbers (isomorphic to Cl0,1), 3D logarithmic functions, due to appearance of two double angle arc tangent functions, allow to include two sets of sheets characterized by discrete coefficients. Formulas for generic and special cases of individual blades and their combinations are provided.

Keywords

• Clifford (geometric) algebra
• multivector logarithm
• computer-aided theory