motionrotation

Motionrotation is a term sometimes used to describe the coordinated movement of a rigid body that combines translation through space with rotation of its orientation. In physics and engineering, rigid-body motion inherently involves both components, and motionrotation can serve as a shorthand for this combined effect in discussions of pose and trajectory.

Mathematically, the pose of a rigid body is described by a rotation R in the special orthogonal group SO(3) and a position vector p in R^3; together they form a homogeneous transformation in SE(3). The instantaneous motion is captured by a twist, consisting of an angular velocity vector ω and a linear velocity vector v. A point with reference position r moves as r' = r + v dt + ω × r dt to first order in small time dt. This framework allows simultaneous accounting for orientation and location changes during motion.

In computer graphics and animation, motionrotation refers to the process of rotating and translating objects to depict movement. Practitioners employ various rotation representations, including rotation matrices, Euler angles, and quaternions; quaternions are often preferred for avoiding gimbal lock and enabling smooth interpolation, such as SLERP, between orientations.

In robotics and mechanical engineering, motionrotation underpins rigid-body kinematics and trajectory planning. Forward kinematics compute the pose from joint parameters, while inverse kinematics determine joint values to achieve a desired pose. Analyses and control tasks frequently use SE(3) and related Lie algebra formalisms to reason about continuous motions in a consistent mathematical framework.

See also SE(3), SO(3), quaternions, SLERP, rigid-body dynamics.