Describing a rotation
A rotation is an isometry that turns an object about a fixed point called the centre of rotation. To describe a rotation completely you need three things: the centre, the angle of turn (for example 90°, 180°, 270°), and the direction (clockwise or anticlockwise).
Every point of the object stays the same distance from the centre after the turn, so the object and image are congruent. The centre of rotation itself does not move — it is an invariant point.
Key idea / 要点
Rotating a point (a, b) about the origin O: 90° anticlockwise → (−b, a); 180° → (−a, −b); 90° clockwise → (b, −a).
Angle and direction matter
A 90° clockwise turn gives a different image from a 90° anticlockwise turn. A 180° turn is the same whether clockwise or anticlockwise.
Worked example / 例题
Rotate point P(4, 1) by 90° anticlockwise about the origin O.
Rule: (a, b) → (−b, a).
So P′ = (−1, 4). The point turns a quarter turn about O, keeping its distance from O the same.
To find the centre of rotation on a grid, draw the perpendicular bisectors of two segments joining points to their images; they meet at the centre. A useful check is that the distance from the centre to a point equals the distance from the centre to its image, because rotation keeps every point the same distance from the centre.
The hands of a clock, a turning wheel and a spinning fan blade are everyday examples of rotation about a fixed centre.
Remember / 记住
- Describe rotation by centre, angle and direction.
- The centre of rotation does not move.
- 180° is the same both ways.
- Object and image are congruent.