Chasles' theorem

Every displacement of a rigid body in three-dimensional space is equivalent to a rotation about an axis combined with a translation along that same axis.

Rigid-body transformation: A transformation that preserves distances between all pairs of points in a body. Chasles' theorem classifies the orientation-preserving rigid-body transformations in three dimensions.
Euclidean isometry: A distance-preserving transformation of Euclidean space. The theorem concerns the orientation-preserving isometries that describe rigid-body displacements.
Michel Chasles: A French mathematician known for foundational work in geometry and the history of mathematics. His name is attached to the theorem describing the axis-and-translation form of rigid displacement.
Robot kinematics: The study of robot motion in terms of joint variables and end-effector positions. Screw coordinates describe joint motions and help compose robot movements.
Pure translation: A rigid-body motion in which every point moves by the same displacement vector without rotation. It is the zero-rotation limiting case of a screw displacement.
Screw axis: The line invariant under a screw displacement, along which translation occurs while the body rotates. It is the axis that makes the rotation and translation share one geometric direction.
Rotation matrix: An orthogonal matrix with determinant one that represents a rotation about the origin. The rotational part of a displacement matrix determines the screw axis direction.
Projective geometry: The study of properties preserved by projection, such as incidence and cross-ratio. Chasles developed influential geometric methods in the nineteenth-century setting from which the theorem emerged.
Product of exponentials formula: A representation of robot pose as a product of matrix exponentials of joint screws. It builds a robot's overall displacement from successive screw motions.
Pure rotation: A rigid-body motion in which points rotate around a fixed axis without translation along it. It is the zero-translation case of the theorem's combined motion.

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Steiner conicRelated: It places the conic construction within a general method for generating algebraic curves.