Editing Rotation matrix (section)

== Uniform random rotation matrices ==
We sometimes need to generate a uniformly distributed random rotation matrix. It seems intuitively clear in two dimensions that this means the rotation angle is uniformly distributed between 0 and 2{{pi}}. That intuition is correct, but does not carry over to higher dimensions. For example, if we decompose {{nowrap|3 × 3}} rotation matrices in axis–angle form, the angle should ''not'' be uniformly distributed; the probability that (the magnitude of) the angle is at most {{mvar|θ}} should be {{math|{{sfrac|1|π}}(''θ'' − sin ''θ'')}}, for {{math|0 ≤ ''θ'' ≤ π}}.

Since {{math|SO(''n'')}} is a connected and locally compact Lie group, we have a simple standard criterion for uniformity, namely that the distribution be unchanged when composed with any arbitrary rotation (a Lie group "translation"). This definition corresponds to what is called ''[[Haar measure]]''. {{Harvtxt|León|Massé|Rivest|2006}} show how to use the Cayley transform to generate and test matrices according to this criterion.

We can also generate a uniform distribution in any dimension using the ''subgroup algorithm'' of {{Harvtxt|Diaconis|Shahshahani|1987}}. This recursively exploits the nested dimensions group structure of {{math|SO(''n'')}}, as follows. Generate a uniform angle and construct a {{nowrap|2 × 2}} rotation matrix. To step from {{math|''n''}} to {{math|''n'' + 1}}, generate a vector {{math|'''v'''}} uniformly distributed on the {{mvar|n}}-sphere {{math|''S''<sup>''n''</sup>}}, embed the {{math|''n'' × ''n''}} matrix in the next larger size with last column {{nowrap|(0, ..., 0, 1)}}, and rotate the larger matrix so the last column becomes {{math|'''v'''}}.

As usual, we have special alternatives for the {{nowrap|3 × 3}} case. Each of these methods begins with three independent random scalars uniformly distributed on the unit interval. {{Harvtxt|Arvo|1992}} takes advantage of the odd dimension to change a [[Householder reflection]] to a rotation by negation, and uses that to aim the axis of a  uniform planar rotation.

Another method uses unit quaternions. Multiplication of rotation matrices is homomorphic to multiplication of quaternions, and multiplication by a unit quaternion rotates the unit sphere. Since the homomorphism is a local [[isometry]], we immediately conclude that to produce a uniform distribution on SO(3) we may use a uniform distribution on {{math|''S''<sup>3</sup>}}. In practice: create a four-element vector where each element is a sampling of a normal distribution. Normalize its length and you have a uniformly sampled random unit quaternion which represents a uniformly sampled random rotation. Note that the aforementioned only applies to rotations in dimension 3. For a generalised idea of quaternions, one should look into [[Geometric algebra#Rotations|Rotors]].

Euler angles can also be used, though not with each angle uniformly distributed ({{Harvnb|Murnaghan|1962}}; {{Harvnb|Miles|1965}}).

For the axis–angle form, the axis is uniformly distributed over the unit sphere of directions, {{math|''S''<sup>2</sup>}}, while the angle has the nonuniform distribution over {{nowrap|[0,{{pi}}]}} noted previously {{Harv|Miles|1965}}.