# Calculation of elements of spin groups using method of averaging in Clifford's geometric algebra

## TL;DR

This paper introduces a generalized averaging method in Clifford's geometric algebra to compute spin group elements in arbitrary dimensions, extending Hestenes' approach for four dimensions.

## Contribution

It develops explicit formulas for spin group elements in any dimension, broadening the applicability of geometric algebra in rotational computations.

## Key findings

- Derived explicit formulas for spin group elements in arbitrary dimensions
- Extended Hestenes' method from 4D to higher dimensions
- Provided computational tools for rotors connecting different frames

## Abstract

We present a method of computing elements of spin groups in the case of arbitrary dimension. This method generalizes Hestenes method for the case of dimension 4. We use the method of averaging in Clifford's geometric algebra previously proposed by the author. We present explicit formulas for elements of spin group that correspond to the elements of orthogonal groups as two-sheeted covering. These formulas allow us to compute rotors, which connect two different frames related by a rotation in geometric algebra of arbitrary dimension.