Exponentials of general multivector (MV) in 3D Clifford algebras

TL;DR

This paper derives closed-form formulas for the exponential of any multivector in 3D Clifford algebras, enabling exact calculations of GA trigonometric and hyperbolic functions with applications in differential equations, signal processing, and robotics.

Contribution

It provides the first explicit closed-form expressions for multivector exponentials in 3D Clifford algebras, facilitating practical computations and applications.

Findings

Derived exact exponential formulas for 3D Clifford multivectors.

Validated formulas against series expansions of GA functions.

Demonstrated applications in solving differential equations and signal processing.

Abstract

Closed form expressions to calculate the exponential of a general multivector (MV) in Clifford geometric algebras (GAs) Cl(p,q) are presented for n=p+q=3. The obtained exponential formulas were applied to find exact GA trigonometric and hyperbolic functions of MV argument. We have verified that the presented exact formulas are in accord with series expansion of MV hyperbolic and trigonometric functions. The exponentials may be applied to solve GA differential equations, in signal and image processing, automatic control and robotics.