Mapping of a Plane onto Itself
What is a Mapping?
A mapping (or transformation) of a plane onto itself is a rule that assigns to every point in the plane exactly one image point that also lies in the same plane. We write:
$f: P \rightarrow P'$
This means the mapping $f$ takes point $P$ and sends it to its image $P'$. The original point $P$ is called the preimage, and the resulting point $P'$ is called the image.
Formal Definition
A mapping of a plane onto itself is a function:
$$f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$$
such that for every point $(x, y)$ in the plane, there exists a unique image point $(x', y')$ also in the plane, where:
$$f(x, y) = (x', y')$$
Key Properties:
- Domain: The entire plane $\mathbb{R}^2$
- Codomain: The same plane $\mathbb{R}^2$
- "Onto" (Surjective): Every point in the plane is the image of some point
- Well-defined: Each point has exactly one image
Types of Plane Mappings
| Transformation | What it does | Formula |
|---|---|---|
| Translation | Slides every point | $(x+a,\; y+b)$ |
| Rotation | Turns around a center | $(x\cos\theta - y\sin\theta,\; x\sin\theta + y\cos\theta)$ |
| Reflection | Flips across a line | $(x,\; -y)$ over $x$-axis |
| Dilation | Scales from a center | $(kx,\; ky)$ |
| Identity | Maps every point to itself | $(x,\; y)$ |