Beeler, M., Gosper, R.W., and Schroeppel, R. HAKMEM. MIT AI Memo 239, Feb. 29, 1972. Retyped and converted to html ('Web browser format) by Henry Baker, April, 1995.


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A quaternion is a 4-tuple which can be regarded as a scalar plus a vector. Quaternions add linearly and multiply (non-commutatively) by
(S1+V1) (S2+V2) = S1 S2 - V1.V2 + S1 V2 + V1 S2 + V1 x V2
where S=scalar part, V=vector part, .=dot product, x=cross product.

If Q = S+V = (Q0,Q1,Q2,Q3), then S = Q0, V = (Q1,Q2,Q3). Define conjugation by (S+V)* = S-V. The (absolute value)^2 of a quaternion is

  2     2     2     2
Q0  + Q1  + Q2  + Q3  = Q Q* = Q* Q .
The non-zero quaternions form a group under multiplication with (1,0,0,0) = 1 as identity and
1    Q*
- = ---- .
Q   Q* Q
The unit quaternions, which lie on a 3-sphere embedded in 4-space, form a subgroup. The mapping F(Q) = P Q (P a unit quaternion) is a rigid rotation in 4-space. This can be verified by expressing P Q as a 4x4 matrix times the 4-vector Q, and then noting that the matrix is orthogonal. F(Q) restricted to the unit quaternions is a rigid rotation of the 3-sphere, and because this mapping is a group translation, it has no fixed point.

We can define a dot product of quaternions as the dot product of 4-vectors. Then Q1.Q2 = 0 iff Q1 is perpendicular to Q2. Let N be a unit vector. To each unit quaternion Q = S+V, attach the quaternion

N Q = -N.V + N S + N x V .
Then it is seen that
(N Q).(N Q) = N.N = 1 and (N Q).Q = 0 .
Geometrically this means that N Q is a continuous unit 4-vector field tangent to the 3-sphere. No such tangent vector field exists for the ordinary 2-sphere. Clearly the 1-sphere has such a vector field.

PROBLEM: For which N-spheres does a continuous unit tangent vector field exist?

Let W be a vector (quaternion with zero scalar part) and Q = S+V. Then

Q W Q* = (S^2 + V.V) W + 2 S V x W + 2 V (V.W) .
Let N be a unit vector and Q the unit quaternion
Q = +-(cos(theta/2) + N sin(theta/2)) .
Q W Q* = (cos theta) W + (sin theta) (N x W) + (1-cos theta) N (N.W) ,
which is W rotated thru angle theta about N. If Q thus induces rotation R, then Q1 Q2 induces rotation R1 R2. So the projective 3-sphere (+Q and -Q identified) is isomorphic to the rotation group (3x3 orthogonal matrices). Projectiveness is unavoidable since a 2 pi rotation about any axis changes Q = 1 continuously into Q = -1.

Let U be a neighborhood of the identity in the rotation group (ordinary 3 dimensional rotations) and U1 the corresponding set of unit quaternions in the neighborhood of 1. If a rotation R carries U into U', then a quaternion corresponding to R carries U1 into U1'. But quaternion multiplication is a rigid rotation of the 3-sphere, so U1 and U1' have equal volume. This shows that in the quaternion representation of the rotation group, the Haar measure is the Lebesgue measure on the 3-sphere.

Every rotation is a rotation by some angle theta about some axis. If rotations are chosen "uniformly", what is the probability distribution of theta? By the above, we choose points uniformly on the 3-sphere (or hemisphere since it is really projective). Going into polar coordinates, one finds

            2      theta 2
P(theta) = -- (sin -----) , 0 < theta < pi .
           pi        2
In particular, the expected value of theta is
pi    2
-- + -- .
2    pi
Quaternions form a convenient 4-parameter representation of rotations, since composition of rotations is done by quaternion multiplication. In contrast, 3-parameter representations like Euler angles or (roll, pitch, yaw) require trigonometry for composition, and orthogonal matrices are 9-parameter. In space guidance systems under development at D-lab, the attitude of the spacecraft is stored in the guidance computer as a quaternion.

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