Transformations Geometry

Transformation geometry is the way to manipulate the shapes of geometry.
Using the terms,
  1. Turn
  2. Flip
  3. Slide
  4. Re-size

Then the object is said to be transformed.
The original shape is called pre-image and the transformed shape is called image. 

In these transformations we have to observe one important thing, that is even though if we apply any of the first three transformations the shape, area, size,lengths remains same. Then those two shapes are said to be concurrent.

But for the fourth transformation that is re-size in that the transformation is said to be similar.but this is not concurrent.

Now we will see different types of transformations we have

  1. Dilation.
  2. Rotations.
  3. Reflections.
  4. Translations.

Dilations

Definition: A Dilation is nothing but a type of transformation which produces an image that is same shape as the original size but it is of different size.

A Dilation stretches and shrinks the original image .

Now we are going to see complete description about Dilations.

This includes following parts..

  1. Scale factor.
  2. Center of Dilations.
  3. Mathematical definition for Dilations.

Scale Factor:

A Dilation of scalar factor k whose center of Dilation is origin can be written as

Dk(x, y) = (kx, ky)

If the scalar factor is greater than one,the image is a stretched one (Enlarged).

k>1 image enlarges

Example Figure:

Dilations

If the Scalar factor is less than one, the image shrinks (reduced).

k<1 image reduces.

If the scalar factor is equal to 1 then those two images are said to be concurrent.

It is possible in the real world but not usually appears.

Center of Dilation:

The center of dilation is a fixed point in a plane.This means the coordinates of point is same in both the planes. This is the only Invariant point in the plane.

Most dilation in the coordinate geometry use (0, 0) as the center of dilation.

Example Figure:

Center of Dilation

here center of dilation is origin.

Rotations

Definition of Rotation:

This is a type of transformation that rotates a figure about a fixed point.

Rotation is also known as Turn.

To rotate a plane figure we need to turn the figure by using a fixed point. This fixed point is called center of Rotation.

here center of this figure indicates center of rotation.

This is another example of rotation

Now we are going to see specifications of Rotations..

  1. Types of Rotations.
  2. Rotation Notation.
  3. Rotation By 90 degrees about the origin. R (origin, 90 degrees).
  4. Rotation By 180 degrees about the origin. R (origin, 180 degrees).
  5. Rotation by 270 degrees about the origin. R (origin, 270 degrees).

Reflections

Definition:

This is a type of transformations,but it flips the shape of the object over the line of reflection.This we can simply says as mirror of the original image. Every reflection has a mirror line. This reflection we consider as a basic types of symmentry. This reflection of a figure in the plane about a line moves its reflected image to where it appears if you viewed using a mirror placed on the line.

Now we are going to see some examples of reflections..

Generally reflections are performed using coordinate notation as they are all on this page.

In mathematical notation we have 3 types of Reflections they are,

  1. Reflection in the X-axis.
  2. Reflection in the Y-axis.
  3. Reflection in the line Y=X.

Reflection in the X-axis:

The general rule for reflection for reflection in x-axis is,

(A , B) is (A , -B).

Now we are going to see some example on reflection in X-axis.

Reflections in the Y-axis:

The general rule for reflection for reflection in Y-axis is,

(A , B) is (-A , B).

Now we are going to see some example on reflection in Y-axis.

Reflections in the line x=y:

The general rule for reflection for reflection in x-axis is,

(A, B) is (B, A).

Now we are going to see some example on reflection in the line Y=X.

Translations

Definition:

This is nothing but simply moving the object without rotating, resizing or any other operation.

We simply just move the object without disturbing its actual dimensions.

Now we are going to see some pictorial examples regarding translations..

Translations

This is the translation of a triangle. They are as follows.

Now we are going to see translations on different types of things..

Generally Translations can be using coordinate axes as they are all on this page.

  1. How To translate?
  2. Translate a point.
  3. Translate an ellipse.
  4. Reflection as translation.

How to Translate:

Generally in translations we can translate by using two types of parameters.

They are:

  1. You can translate either by using Angle and Distance.
  1. Or You can translate by using coordinates X and Y.

Now we will see how these both will work.

Angle and Distance:

In this we can translate the given shape by using angle in degrees and the distance in units now we are going to see this angle and distance translation in the form of an example.


  Example:

Given triangle translate the triangle in terms of angle 30 degrees and distance 15 units.

Solution:
This translation can be done here as follows.

Angle and Distance

Translation using coordinates:

In this translation can be done using coordinates means moving the direction of

the image with respect to coordinates in x and y.

This can be shown in example as follows..


Example:

Translate the given triangle with 30 units in the X direction and 40 units in the Y direction.

Solution:

This can be translates as follows.

Translation using Coordinates

Now the given point translates from (x , y) $\rightarrow$ (x+30, y+40)

Topics in Transformations Geometry