## 2D transformation

**Transformation**means that changing some graphics into different we've various kinds of

**transformation**like

**translation**,

**scaling**,

**up or down**,

**rotation**,

**shearing**. when a

**transformation**text place on a

**2D**plane is called

**2D transformation**.

### Homogeneous coordinates:

To perform a sequence of transformation like translation followed by rotation and

**scaling**we'd like to follow a sequential process.

1) translate the coordinate.

2)

**rotate**the

**translated**coordinate.

3)

**scale**the

**rotated co-ordinate**to complete composite

**transformation**.

For this process we have to use 3*3 transformation Matrix Institute of 2*2

**transformation matrix**. To convert a 2*2 Matrix into 3*3 we add an extra co-ordinate 'w'.

In this way beacon represent the point by 3 e number instead of two number which is called

**homogeneous co-ordinate**system.

### Translation:

Translation moves ad object to a different position on the screen. You can translate a point in**2 dimensional**by adding

**translation coordinate**. (tx,ty) to the original co-ordinate (x, y) to get the new coordinate (x',y').

From the above figure we can write

x' = x + tx

y' = y + ty

The pear TX, TY is called the translation vector or Swift vector the above equation can also be represented by using the column vector

P=[x][y]P'

That is P' = P+T

### Rotation:

Invitation process be rotate the object at particular ∠ θ from its origin from the figure we can see that the point P(x,y) is the located at ∠ φ from the from the horizontal x co-ordinate with distance ‘ r ’.from the origin let us suppose you want to rotate it at ∠ θ and after

**rotating**it to a new location you will get new point 'P'(x',y')

Using the standard of Krishna Matric rules the original coordinates of point P(x,y) can be represent as

X = r cosφ (1)

Y = r sinφ (2)

Same way we are able to represent the point P'(x',y') as

x' = rcos(φ+θ)

y' = rsin(φ+θ)

x' = r cos(φ + θ)= r cosφ cosθ - r sinφ sinθ (3)

y' = r sin(φ + θ)= r cosφ sinθ + r sinφ cosθ (4)

Substitute the value of equation (1) and (2) in equation (3) and (4)

x' = x cosθ – y sinθ (5)

y' = x sinθ + y cosθ (6)

Representing the above equation in matrix form [x' y'] = [x y][cosθsinθ-sinθcosθ]

Or P' = P.R

Where R is the

**rotation**matrix.

### Scaling:

**Scaling transformation**is used to change the size of an object in the

**scaling**process either we can expand of compress the dimension of the object.

**Scaling**can be achieved by multiplying the original coordinates of the object with the spelling factor to get the desired result. Let us assume that the original co-ordinate are (x, y) and the

**scaling**factor are (Sx,Sy) and the produce coordinate are (x',y') this can be mathematically represented as

x'= x.Sx. and y'=y.Sy.

Sx and Sy is the

**scaling**factor with the respect to x & y co-ordinate.

The above equation can also be represented in matrix from as [x' y'] = [x y][Sx Sy]

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