Euler's rotation theorem

Every proper rotation of a rigid body about a fixed point is equivalent to a single rotation about an axis through that point.

Rotation matrix: A square matrix that represents a linear transformation of coordinates, including rotations. A proper rotation is represented by an orthogonal matrix with determinant one.
Rigid body: An idealized object whose points maintain fixed distances from one another. The theorem describes the net orientation change of such an object around a fixed point.
Leonhard Euler: An eighteenth-century Swiss mathematician whose work shaped analysis, mechanics, and geometry. Euler established the result that bears his name.
Screw theory: A framework representing rigid-body displacement as rotation about and translation along an axis. Unlike the theorem's fixed-point setting, general rigid displacement can include translation along the axis.
Orthogonal matrix: A real square matrix whose transpose is its inverse, preserving lengths and angles. Rotation matrices are orthogonal, a property that constrains their eigenvalues.
Euclidean transformation: A transformation of space that preserves distances. Proper rotations are distance-preserving transformations with a fixed point and no reflection.
Rigid-body dynamics: The study of the motion of bodies whose internal distances remain fixed. The theorem provides a geometric foundation for describing changes in rigid-body orientation.
Chasles' theorem: The result that any proper rigid displacement in three dimensions is a rotation about an axis combined with translation along it. It generalizes the fixed-point rotation result to rigid motions that need not keep a point fixed.
Eigenvalue: A scalar by which a linear transformation scales a corresponding nonzero vector. Every three-dimensional proper rotation has eigenvalue one, yielding an unchanged axis direction.
Proper rotation: A rotation that preserves orientation, represented by an orthogonal transformation with determinant one. The theorem applies to proper rotations, not reflections or other orientation-reversing motions.

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