Quaternions 四元数

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In lecture, we saw that we could represent rotations as rotation matrices, or other angles in the axis angle representation(轴角参数化). In this segment, we'll discuss a final method of representation for rotations, namely using quaternions(四数).


A quaternion is a fourtuple. This set of numbers is often interpreted as a constant component q_zero, and a three-dimensional vector component q.

Operations with Quaternions

Application of Quaternions

The angle of rotation is a scalar, and the axis of rotation is a three-dimensional vector.

Rearrange the form, we obtained:

For its beautiful properties, we can easily compose two and more rotations:

Advantages of Quaternions

Euler angle have singularities in two poles, the quarternions can solve the problem. Rotation matrix have 9 indentical terms which is complex that using quaternions.