Rotation
Geometry & MeasurementA rotation turns a figure by a specified angle about a fixed center point, producing a congruent image.
Formula
(x,y) \to (x\cos t - y\sin t, \ x\sin t + y\cos t)
Definition
A rotation turns a shape around a fixed point called the center of rotation. You specify how many degrees to turn and in which direction (clockwise or counterclockwise). The shape's size and angles stay the same.
Example
Rotating a square $90$ degrees counterclockwise about its center: the square looks the same because it has $4$-fold rotational symmetry. Rotating a scalene triangle $90$ degrees about the origin moves each vertex to a new location.
Key Insight
When we say "rotate $180$ degrees," it does not matter if you go clockwise or counterclockwise because you end up at the same place. But $90$ degrees clockwise is different from $90$ degrees counterclockwise.
Definition
A rotation by angle $\theta$ counterclockwise about the origin maps $(x, y)$ to $(x\cos\theta - y\sin\theta, \ x\sin\theta + y\cos\theta)$. For rotation about an arbitrary center $(h, k)$, first translate so $(h,k)$ is at origin, rotate, then translate back. Rotations are orientation-preserving isometries.
Example
Rotate point $(3, 0)$ by $90$ degrees counterclockwise: $(3\cos(90^\circ) - 0\sin(90^\circ), 3\sin(90^\circ) + 0\cos(90^\circ)) = (0, 3)$. Rotate $(0, 3)$ by $90$ degrees: $(-3, 0)$. Rotate $(-3, 0)$: $(0, -3)$. Four $90$-degree rotations return to start.
Key Insight
A rotation by angle $\theta$ and a rotation by $\theta+360^\circ$ are identical. This is why rotational symmetry is described by the smallest rotation that maps a shape to itself. For a regular n-gon, the minimum rotation is $360/n$ degrees.
Definition
Rotation by $\theta$ about the origin is the linear map $R(\theta) = \begin{bmatrix}\cos\theta & -\sin\theta\\\sin\theta & \cos\theta\end{bmatrix}$, an element of the special orthogonal group $SO(2)$. Composing rotations: $R(\theta_1)R(\theta_2) = R(\theta_1+\theta_2)$ ($SO(2)$ is abelian). In 3-D, rotations form $SO(3)$, which is non-abelian (order of rotations about different axes matters).
Example
In 3-D, rotating $90$ degrees about the x-axis then $90$ degrees about the y-axis gives a different result than the same rotations in reverse order: the non-commutativity of $SO(3)$ is fundamental in robotics (gimbal lock) and quantum mechanics (spin operators).
Key Insight
The non-commutativity of 3-D rotations ($SO(3)$) is a deep geometric fact with profound physical consequences: it underlies the non-commutativity of quantum angular momentum operators $[L_x, L_y] = i\hbar L_z$, connecting geometry directly to the quantum mechanical structure of atoms.