Kiertopiste

Kiertopiste is a term used in Finnish geometry to denote the center of rotation. In two-dimensional space, a rotation is a rigid motion that turns every point around a fixed point by a given angle, and the kiertopiste is that fixed point.

In a nontrivial rotation, the kiertopiste O remains fixed while every other point P moves to a new position P′ such that the distance OP equals OP′ and the angle between OP and OP′ equals the rotation angle θ. The set of points at a constant distance from O forms circles, and during a rotation each point travels along its circle around O.

Explicit coordinates can describe a rotation about O by angle θ. If P is represented by (x, y) and the kiertopiste is the origin, P′ = (x cos θ − y sin θ, x sin θ + y cos θ). For a rotation around a general kiertopiste O = (a, b), P′ = (a + (x − a) cos θ − (y − b) sin θ, b + (x − a) sin θ + (y − b) cos θ).

In three dimensions, rotations are typically around an axis rather than a single point, so the concept of a kiertopiste is mainly used for planar rotations. The term emphasizes the fixed point that defines the rotation’s center in a plane.

Applications of the kiertopiste concept appear in geometry problems, computer graphics, robotics, and animation, where rotations are fundamental transformations. See also center of rotation, pivot, rotation matrix, and rigid body motion.