Rotationtotranslation

Rotationtotranslation is a concept used to represent a three-dimensional rotation as a rigid-body motion that includes a translation component. In this view, a rotation R in SO(3) about an axis that may not pass through the origin is expressed as a pose T = [R, t] in SE(3), where the translation component t depends on a chosen anchor point c on the rotation axis. The standard relation is t = (I − R) c, so the action on a point x is x' = R x + t. This embedding allows one to treat certain rotations as combined rotation and translation moves in a single homogeneous transformation.

If the rotation axis passes through the origin (c = 0), the translation vanishes and the motion is a pure rotation. For an offset axis, the translation encodes the offset and yields the same R when applied to points, but with a nonzero translational part. Importantly, the mapping from (R, c) to t is not canonical: different choices of c on the same axis yield different t values that satisfy t = (I − R) c. This non-uniqueness means rotationtotranslation depends on a chosen anchor and is most useful when the axis location is known or constrained.

Applications and relationships: rotationtotranslation is employed in robotics, computer graphics, and computer vision to unify rotation and translation in pose representations, simplify interpolation of rigid motions, and facilitate optimization over SE(3) rather than SO(3) alone. It is closely related to screw theory and the exponential map between SE(3) and its Lie algebra se(3), providing intuition for how rotational motion can be paired with translational components.

See also: SE(3), SO(3), axis-angle representation, screw theory, exponential map, pose estimation.