Rotating a Unit Vector in 3D Using Quaternions

start position



u

1

(red)

θ

1

2π

3

ϕ

1

0

final position



u

2

(green)

θ

2

π

2

ϕ

2

2π

3

rotating



u

1

around OverscriptBox[\(\*SubscriptBox[\(u\), \(1\)]\( \)\), \(\)]



u

2

(blue)



u

1

to



u

2

orbit

α

reset α

ImageSize

size

350

A quaternion is a vector in

4



with a noncommutative product (see [1] or Quaternion). Quaternions, also called hypercomplex numbers, were invented by William Rowan Hamilton in 1843. A quaternion can be written

q=w+xi+yj+zk

or, more compactly,

(w,x,y,z)

or

w,



r



, where the noncommuting unit quaternions obey the relations

2

i

=

2

j

=

2

k

=ijk=-1

.

A quaternion can represent a rotation axis, as well as a rotation about that axis. Instead of turning an object through a series of successive rotations using rotation matrices, quaternions can directly rotate an object around an arbitrary axis (here



u

1





u

2

) and at any angle

α

. This Demonstration uses the quaternion rotation formula

p

1

'=q

p

1

-1

q

with

p

1

=0,



u

1



, a pure quaternion (with real part zero),

q=cos

α

2

,



u

3

sin

α

2

, normalized axis



u

3

=



u

1





u

2





u

1

·



u

2

, and for a unit quaternion,

-1

q

=

-1

q

=

-

q

, where the quaternion conjugate for

p=(a,b,c,d)

is

-

p

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

.