a rotation

A rotation is a turn about a fixed pivot. Picture a clock hand sweeping around its central pin, or a merry-go-round spinning about its axle: every point swings along a circular arc, keeping its distance from the pivot constant, while the pivot itself stays put. To describe one you need exactly two ingredients — where the pivot is (the centre of rotation) and how far around to turn (the angle, with a sign telling clockwise from counterclockwise).

Precisely, a rotation about a centre O through a signed angle theta sends each point P to the point P' such that |OP'| = |OP| and the directed angle from OP to OP' equals theta. By convention a positive theta turns counterclockwise. The centre O is the one and only fixed point (for any theta that is not a whole number of full turns). A rotation preserves distance and orientation, so it is a direct isometry — a clockwise-labelled triangle stays clockwise. A rotation by 180 degrees has a special name, a point reflection or half-turn: it sends each P to the point directly opposite O at equal distance, i.e. (x, y) to (2h - x, 2k - y) about centre (h, k).

Composition reveals a clean structure: two rotations about the same centre simply add their angles (turn by alpha, then beta, and you have turned by alpha + beta). Two rotations about different centres compose to another rotation whose angle is the sum — unless the angles cancel to a whole turn, in which case the result is a translation, a genuinely surprising fact. Rotations are the heart of rotational symmetry: a figure has n-fold rotational symmetry when a turn of 360/n degrees about its centre lands it exactly on itself, which is why a square (n = 4) and a regular hexagon (n = 6) look unchanged after the right partial turn.