# Quaternions in General

This page describes general definitions and manipulations with quaternions while the application of quaternions to define the orientation of rigid bodies is describes in QuaternionOrientation.

### Definition

Quaternions are an expansion of compex numbers. A quaternion has three imaginary elements: , and and can be written in the form:

• Where , and are independent imaginary numbers with the relation:

• Further relations hold:

•   For better readablity a quaternion can also be written in matrix notation:

• where the basis can be omitted.

A quaternion with no real component is called a pure quaternion. According to the notation of coordinates of a 3D vector, pure quaternions are marked by an underline:

• ### Polar Form

In correspondance with complex numbers quaternions can also be written in the polar form:

• where has to be normalized. For an interpretation of and see Axis Angles. is the norm of the quaternion .

The polar form of a unit quaternion ( ) can be expanded and written in power series:

• Since the square of a normalized pure quaternion is , the power series can be rewritten to:

• Sorting the series for real and imaginary parts yield

• where the power series of the real part is equal to the power series of and the power series of the imaginary part is equal to the power series of . So the following relation holds:

• ### Math

Multiplication of two quaternions:

• Exponentiation, logarithm of a quaternion

• is computed as

•  The exponential of a pure quaternion simplifies to:

• And the logarithm of a normalized quaternion simplifies to:

• Quaternions (last edited 2010-09-18 10:01:29 by StefanEngelke)