# 2D Transformation in Computer Graphics Solved Examples

__2D Transformation in Computer Graphics:__

## Translation:

Let us imagine a point P in a 2D plane. Assume that P’s co-ordinate (x,y) is depict the current position.

Now, if we force P to move Δx distance horizontally and at the same time Δy distance vertically then the changed location of P becomes **(x+Δx, y+Δy)**.

In terms of object transformation, we can say that the original point object p(x,y) translates to becomes P'(x’,y’) and the amount of translation apply is the vector.

Algebraically,

**x’=x+Δx
y’=y+Δy**

## Rotation:

This transformation is used to rotate the objects about any point in a reference frame. Unlike translation, rotation brings about changes in position as well as orientation. The point about which the object rotates, it says the Pivot point or Rotation point.

**Rotation about Origin :**

Consider a trial case where the pivot point is the origin as shown in Figure:

The rotating point P(x,y) cab represents as –

**x=rcosϕ, y=rsinϕ**

where (r, ϕ) is the polar co-ordinate of P. When this point P is rotated through an angle θ in the anti-clockwise direction, the new point P'(x’,y’) becomes,

**x’=rcos(θ+ϕ) y’=rsin(θ+ϕ)**

## Rotation about an Arbitrary Pivot Point:

The pivot point is an arbitrary point P_{p} having coordinates (x_{p}, y_{p}).After rotating P(x,y) through a positive θ angle its new location is x’y'(P’).

## Scalling:

Scaling is a transformation that changes the size or shape of an object. Scaling origin can be carried out by multiplying the coordinate values (x,y) of each vertex of a polygon, or each endpoint of a line or the centre point and peripheral definition points of closed curves like a circle by scaling factors s_{x} and s_{y} respectively to produce the coordinate (x’,y’).

The mathematical expression for pure scaling is :

## Homogeneous Coordinates:

The **Homogeneous Coordinate** is a method to perform certain standard operations on points in Euclidean space that means of matrix multiplications. Normally, we add a coordinate to the end of the list and make it equal to 1. Thus the two-dimensional point (**x,y**) becomes(**x,y,1**) in homogeneous coordinates, and the three-dimensional point (**x,y,z**) becomes (**x,y,z,1**)

Homogeneous Coordinates are not Euclidean coordinates, they are the natural coordinates of a different type of geometry is say Projective Geometry.

## Coordinate Transformations:

Coordinate Transformations Geometric Transformation of 2D objects which are well-defined with respect to a **Global Coordinate System**, that is say the **World Coordinate System (WCS)**. It is often found convenient to define quantities with respect to a Local Coordinate System that is also called **Model Coordinate System** or **User Coordinate System (UCS)**.

## Affine Transformation:

An affine transformation involving only translation, rotation and reflection preserves the length and angle between two lines. All two-dimensional transformation where each of the transformed coordinates x’ and y’ is a linear function of the original coordinates x & y as:

**x’=A**

y’=A

_{1}x+B_{1}y+C_{1}y’=A

_{2}x+B_{2}y+C_{2}where A

_{1}, B

_{1}, C

_{1}are parameters fixed for a given transformation type.

__2D Transformation Solved Examples:__

2D Transformation in Computer Graphics solved problem is given below: