Geometry is an essential part of mathematics. It is not just the study of shapes. It is a vast field of study. There are many different branches of geometry.
Few important of them are the following:
1) Euclidean Geometry
2) Non-Euclidean Geometry
3) Plane (2D) Geometry
4) Solid (3D) Geometry
5) Differential Geometry
6) Coordinate Geometry
7) Descriptive Geometry
8) Transformation Geometry
The last one "Transformation Geometry" is an important branch of geometry which studies about the transformation of geometrical figures. We shall discuss transformation geometry in this page below.

## Transformation Geometry Definition

As the name "transformation geometry" itself suggests that it is a branch of geometry which studies about transformation of geometric shapes. It is also termed as "transformational geometry".
Sometimes, while analyzing and solving problems about different geometrical figures, one needs to transform them to different shapes, different orientations, different sizes or different dimensions.
All of this is included under transformation geometry. The transformation of shapes is also known as geometric transformation.

## How to do a Transformation in Geometry

In geometry, transformation can be accomplished in many ways which are as follows:
1) Translation
In order to perform this type of transformation, each and every point of an image must be shifted by a particular distance in a particular direction. Translation is actually the "shifting" of whole shape.
The dimensions of the image must be unchanged as shown in the following diagram:

2) Rotation
In rotational transformation, a figure has to be rotated either clockwise or anticlockwise assuming a point (either exterior or interior to the figure) as a center. Each point of image is to be shifted with a particular angle.
Rotation is shown below:

3) Reflection
This type of transformation is same as the mirror reflection. In order to perform this, the image has to be flipped around an imaginary line or axis, keeping each point at a same distance on both sides of the line.
Reflection is illustrated by the diagram below:

4) Resizing
Resizing includes the act of making a geometrical figure either smaller or bigger in the same proportion. Resizing may include dilation, enlargement, compression or contraction.
The diagram shown below illustrates resizing more clearly.

## Transformation Geometry  Rules

In order to perform, different types of transformations, we need to shift the coordinates of almost each point of an image.
There are certain rules for the transformation of coordinates. These rules are as follows:
Rules for Translation
T$_{a,b}$ : (x, y) $\Rightarrow$ (x + a, y + b)

Rules for Rotation
Clockwise 90$^{\circ}$ : (x, y) $\Rightarrow$ (y, - x)
Anticlockwise 90$^{\circ}$ : (x, y) $\Rightarrow$ (- y, x)
Clockwise or anticlockwise 180$^{\circ}$ : (x, y) $\Rightarrow$ (- x, - y)

Rules for Reflection
1) Point Reflection
About origin : (x, y) $\Rightarrow$ (- x, - y)

2) Line Reflection
About X axis : (x, y) $\Rightarrow$ (x, - y)
About Y axis : (x, y) $\Rightarrow$ (- x, - y)
About line y = x : (x, y) $\Rightarrow$ (y, x)
About line y = - x : (x, y) $\Rightarrow$ (- y, - x)

Rules for Resizing
1) Enlargement
By factor "p" : (x, y) $\Rightarrow$ (p x, p y)

2) Contraction

By factor "p" : (x, y) $\Rightarrow$ ($\frac{x}{p}$, $\frac{y}{p}$)

## Transformation Geometry Problems

Few problems based on transformation geometry are given below:
Problem 1: A triangle whose vertices are (0, 4), (0, 0) and (6, 0) is enlarged to reach the vertices at (0 ,10), (0, 0) and (15, 0) respectively. Calculate the factor by which the triangle is enlarged.
Solution: Let the factor be k. For translation by factor "k", we have
(x, y) $\Rightarrow$ (k x, k y)
Here, (0, 4) $\Rightarrow$ (0, 10)
and (6, 0) $\Rightarrow$ (15, 0)
4 = 10 k

k = $\frac{10}{4}$ = 2.5
Also

15 = 6 k

k = $\frac{15}{6}$ = 2.5

Hence, the transformation factor is 2.5 unit.

Problem 2: What would be the coordinates of point (7, -9), under the translation T$_{1, 2}$?
Solution: For translation, we have
T$_{a,b}$ : (x, y) $\Rightarrow$ (x + a, y + b)
Therefore,
T$_{1, 2}$ : (7, - 9) $\Rightarrow$ (7 + 1, -9 + 2)
So, (7, - 9) $\Rightarrow$ (8, - 7).

Problem 3: Under reflection about X axis, what would be the coordinates of point (4, 5)?
Solution: For reflection, we know that
About X axis : (x, y) $\Rightarrow$ (x, - y)
Therefore,
(4, 5) $\Rightarrow$ (4, - 5).