A Universal Exponential Law for Real Alternative Algebras and its Geometric Applications

Euler's identity $e^{i\theta}=\cos\theta+i\sin\theta$ and De~Moivre's formula

describe rotations generated by an imaginary unit $i$ satisfying $i^2=-1$.

In associative and alternative hypercomplex algebras, however, arbitrary elements

$A$ need not square to $-1$, the exponential map may lose geometric meaning, and

fractional powers $A^x$ often lack closed forms.

 

This paper systematizes existing exponential formulas into a unified algorithmic framework with computational advantages.F

admits a two-term Euler-type fractional power: \emph{$A^x$ has a closed form if and

only if $A^2$ is a nonzero real scalar}.  When $A^2<0$ we obtain the elliptic

(rotation) formula with $ \alpha^x= (\sqrt{\lvert A^{2} \rvert})^{x} \in \mathbb{R}_{>0} $

\[

A^x = \alpha^x\!\left[\cos\Bigl(\tfrac{\pi x}{2}\Bigr)

+ E\sin\Bigl(\tfrac{\pi x}{2}\Bigr)\right],

\qquad E=\frac{A}{\sqrt{-A^2}},\; E^2=-1,

\]

extending Euler and De~Moivre to all alternative algebras, including quaternions

and octonions.  When $A^2>0$, we obtain the symmetric hyperbolic (boost) formula

\[

A^x = \alpha^x\!\left[\cosh\Bigl(\tfrac{\pi x}{2}\Bigr)

+ H\sinh\Bigl(\tfrac{\pi x}{2}\Bigr)\right],

\qquad H=\frac{A}{\sqrt{A^2}},\; H^2=+1,

\]

which covers split-quaternions and other hyperbolic directions.

 

Both formulas follow from the principal logarithm

$\log(A)=A\pi/2$ on the generated $1$--$A$ plane and remain valid for all real

fractional exponents.  A collapse identity is proved:

\[

A^{A x}= \exp(A^2\pi x/2),

\]

giving the classical $i^i=e^{-\pi/2}$ and its hyperbolic analogues as special

cases.  When $A^2\notin\mathbb{R}$ or $A$ is a zero divisor (as in the sedenions),

the two-term formulas fail exactly, giving a precise obstruction.

 

This yields an algorithmic framework for fractional exponentiation in all

Cayley--Dickson and split algebras: test $A^2$, normalize, and apply the elliptic

or hyperbolic closed form when allowed.  Applications include hypercomplex

cyclotomy, Galois actions along arbitrary imaginary directions, and rotation/boost

operators in higher dimensions.  All claims are supported by independent numerical

validation, with reproducible code provided.

 

 

Critically, we demonstrate that this algebraic approach eliminates the need for the computationally expensive \texttt{arctan2} function required by standard polar decomposition methods. Benchmarking against standard library implementations reveals a mean computational speedup of approximately $2x+$ times faster for quaternion roots than SciPy Rotation as standard library, attributed to the replacement of transcendental inverse trigonometric functions with hardware-accelerated algebraic operations. Numerical validation confirms that this performance gain incurs no penalty in accuracy, maintaining machine precision ($10^{-16}$) and exhibiting superior stability near the identity element. This method offers a unified, high-performance primitive for geometric rotations in computer graphics, robotics, and hypercomplex neural networks.