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3D graphics eBook - Course Materials Repository

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Quaternions and spatial rotation 132<br />

Example<br />

The conjugation operation<br />

A rotation of 120° around the first diagonal permutes i, j, and k<br />

cyclically.<br />

Consider the rotation f around the axis , with a rotation angle of 120°, or 2π ⁄ 3 radians.<br />

The length of is √3, the half angle is π ⁄ 3 (60°) with cosine ½, (cos 60° = 0.5) and sine √3 ⁄ 2 , (sin 60° ≈ 0.866). We<br />

are therefore dealing with a conjugation by the unit quaternion<br />

If f is the rotation function,<br />

It can be proved that the inverse of a unit quaternion is obtained simply by changing the sign of its imaginary<br />

components. As a consequence,<br />

and<br />

This can be simplified, using the ordinary rules for quaternion arithmetic, to<br />

As expected, the rotation corresponds to keeping a cube held fixed at one point, and rotating it 120° about the long<br />

diagonal through the fixed point (observe how the three axes are permuted cyclically).

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