# Geometric Transformations

A **geometric transformation** is a one-to-one correspondence between every point in a plane (or space) and another point in the same plane (or space).

There are various types of geometric transformations: rotations, translations, isometries, and so on.

Formally, a geometric transformation is any invertible application T that maps each point of a non-empty set X to a point P' in set Y.

$$ T: X \rightarrow Y $$

Where set Y is the image T(X) of set X under the application T.

In simpler terms, geometric transformations are mathematical operations that change the position or shape of an object in a geometric space. Transformations can be applied to various geometric entities such as points, lines, shapes, and solid figures.

## An Example

Let's consider a geometric figure ABC.

If I rotate a geometric figure by a certain angle alpha, I get a one-to-one correspondence between points A and A', B and B', C and C'.

The points A', B', and C' after the transformation are called **images** or **transformed points** of points A, B, and C, respectively.

Considering the rotation as a bijective function f, we can also write:

$$ A' = f(A) $$

$$ B' = f(B) $$

$$ C' = f(C) $$

This notation reads as "*A' is equal to f of A*", and so on.

And of course, the same applies to all other points and segments of the figure.

For example, the segment A'B' is the image of segment AB.

This is a practical example of a geometric transformation.

**Note**. In this example, point P has an image that corresponds to itself. In other words, point P does not change its position in the plane after the rotation. In this case, P is called a fixed point.

## Types of Geometric Transformations

There are several types of geometric transformations.

Here is a list of the main geometric transformations:

**Translation**

Moving all points of a figure a constant distance in a specified direction. The translated figure retains the same shape and size as the original.

**Rotation**

Circular movement of a figure around a fixed point (the center of rotation). The distance of each point in the figure from the center of rotation remains constant.

**Reflection (or Mirroring)**

Flipping a figure across a line (the axis of reflection), creating a mirror image.

**Isometry**

Isometries are geometric transformations that preserve distances between points and the dimensions of objects. For example, translation, rotation, and reflection are isometric transformations.**Homothety (Dilation or Scaling)**This transformation does not preserve dimensions but maintains the proportions and overall shapes of objects. It changes the size of a figure by multiplying the distances of all its points from a fixed point (the center of dilation) by a constant factor.

## Observations

Some observations and side notes:

- Geometric transformations can be represented mathematically using equations, matrices, or functions. For example, a translation can be expressed by adding a constant vector to each point, while a rotation can be represented by a rotation matrix. And so on.

And so on.