Difference between revisions of "Pure and unit quaternions"
From Robotics
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A quaternion whose vector part is zero equals a real number corresponding to the scalar part. | A quaternion whose vector part is zero equals a real number corresponding to the scalar part. | ||
− | A quaternion whose scalar part is zero, is called a | + | A quaternion whose scalar part is zero, is called a pure quaternion: |
− | |||
:<math> | :<math> | ||
− | ( | + | Im(q) = 0 |
</math> | </math> | ||
− | + | The square of pure quaternions is always real and not positive: | |
:<math> | :<math> | ||
− | \ | + | q^2 \in \R, q^2\le 0 |
− | + | </math> | |
− | + | ||
− | + | Multiplication of pure quaternions leads to the following simplified equation (for the general equation see chapter [[Multiplication of quaternions]]): | |
− | + | :<math> | |
+ | (0,\vec{\mathbf{x}})(0,\vec{\mathbf{y}})=(-\vec{\mathbf{x}}\cdot\vec{\mathbf{y}},\vec{\mathbf{x}}\times\vec{\mathbf{y}}) | ||
+ | </math> | ||
+ | |||
+ | ===Unit quaternion=== | ||
+ | A unit quaternion, also called normalized quaternion, has a magnitude of 1: | ||
+ | :<math> | ||
+ | |q| = q_0^2 + q_1^2 + q_2^2 + q_3^2 = 1 | ||
+ | </math> | ||
+ | A unit quaternion can be created from any quaternion by dividing it and so the four components by its norm: | ||
+ | :<math> | ||
+ | q_u = \frac{q}{|q|} = \frac{q_0}{|q|} + \frac{q_1}{|q|}i + \frac{q_2}{|q|}j + \frac{q_3}{|q|}k | ||
+ | </math> | ||
+ | The product of two unit quaternions and the inverse of a unit quaternion are again unit quaternions. | ||
+ | |||
+ | If <math>q</math> is a unit quaternion, its [[Basic properties of quaternions|inverse]] equals its [[Basic properties of quaternions|conjugate]]: | ||
+ | :<math> | ||
+ | q^{-1}=q^* | ||
</math> | </math> |
Revision as of 15:45, 23 June 2015
← Back: Basic properties of quaternions | Overview: Quaternions | Next: Addition of quaternions → |
Pure quaternion
A quaternion whose vector part is zero equals a real number corresponding to the scalar part.
A quaternion whose scalar part is zero, is called a pure quaternion:
The square of pure quaternions is always real and not positive:
Multiplication of pure quaternions leads to the following simplified equation (for the general equation see chapter Multiplication of quaternions):
Unit quaternion
A unit quaternion, also called normalized quaternion, has a magnitude of 1:
A unit quaternion can be created from any quaternion by dividing it and so the four components by its norm:
The product of two unit quaternions and the inverse of a unit quaternion are again unit quaternions.
If is a unit quaternion, its inverse equals its conjugate: