Rotating a Cube Using Quaternions

orientation of



u

1

(black vector)

θ

1

1.57

ϕ

1

0

rotating around



u

1

α

reset α

opacity

0.8

A quaternion can represent both a rotation axis and the angle of rotation about this axis (a vector and a scalar). Instead of turning an object through a series of successive rotations with rotation matrices, quaternions are used to rotate an object more smoothly around an arbitrary axis (here



u

1

) and at any angle. This program uses the quaternion rotation formula:

p

1

'=q

p

1

-1

q

with

p

1

=0,



u

1



(a pure quaternion),

q=cos(α/2),



u

1

sin(α/2)

, and for a unit quaternion

-1

q

=

-

q

, such that if

q=(a,b,c,d)

,

-

q

=(a,-b,-c,-d)

. A derivation of these formulas is given in [1].