Editing Quaternion (section)

===Geodesic norm===
The [[Great-circle distance|geodesic distance]] {{nowrap|{{math|''d''<sub>g</sub>(''p'', ''q'')}}}} between unit quaternions {{mvar|p}} and {{mvar|q}} is defined as:<ref name=Geodesic>{{cite journal |last1=Park |first1=F.C. |last2=Ravani |first2=Bahram |title=Smooth invariant interpolation of rotations |journal=ACM Transactions on Graphics |date=1997 |volume=16 |number=3 |pages=277–295 |doi=10.1145/256157.256160|s2cid=6192031 |doi-access=free }}</ref>

<math display=block>d_\text{g}(p, q) = \lVert \ln(p^{-1} q) \rVert.</math>

and amounts to the absolute value of half the angle subtended by {{mvar|p}} and {{mvar|q}} along a [[Arc (geometry)|great arc]] of the {{math|S<sup>3</sup>}} sphere.
This angle can also be computed from the quaternion [[dot product]] without the logarithm as:

<math display=block>d_\text{g}(p, q) = \arccos(2(p \cdot q)^2 - 1).</math>