Question

Rotational transformation

Answer 1

Rotational transformation refers to the operation that rotates points in a coordinate system about a specified point, usually the origin (0,0). This type of transformation is widely used in mathematics, physics, computer graphics, and engineering.

### 1. **Mathematical Representation**:

In a two-dimensional Cartesian coordinate system, a rotation can be represented using matrices. If you have a point \( P(x, y) \), the rotation of this point by an angle \( \theta \) about the origin can be expressed using the following matrix:

\[
R(\theta) =
\begin{bmatrix}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)
\end{bmatrix}
\]

The new coordinates \( P'(x', y') \) after the rotation can be calculated by multiplying the rotation matrix by the point vector:

\[
\begin{bmatrix}
x' \\
y'
\end{bmatrix}
=
R(\theta) \cdot
\begin{bmatrix}
x \\
y
\end{bmatrix}
=
\begin{bmatrix}
x\cos(\theta) - y\sin(\theta) \\
x\sin(\theta) + y\cos(\theta)
\end{bmatrix}
\]

### 2. **Rotation About a Point Other Than the Origin**:

If you wish to rotate a point around another point \( Q(h, k) \), the procedure involves translating the point such that \( Q \) becomes the origin, applying the rotation, and then translating back.

#### Steps:
1. Translate the point \( P(x, y) \) so that \( Q(h, k) \) is at the origin:
\[
P' = (x - h, y - k)
\]

2. Apply the rotation:
\[
P'' = R(\theta) \cdot P' =
\begin{bmatrix}
(x - h)\cos(\theta) - (y - k)\sin(\theta) \\
(x - h)\sin(\theta) + (y - k)\cos(\theta)
\end{bmatrix}
\]

3. Translate back:
\[
P_{\text{final}}