# 2D Transformation in Computer Graphics Solved Examples

## Translation:

Let us imagine a point P in a 2D plane. Assume that P’s coordinate (x,y) depicts 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.

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

The rotating point P(x,y) represents as –
x=rcosϕ, y=rsinϕ
where (r, ϕ) is the polar coordinate 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 Pp having coordinates (xp, yp). 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, each endpoint of a line or the center point and peripheral definition points of closed curves like a circle by scaling factors sx and sy respectively to produce the coordinate (x’,y’).
The mathematical expression for pure scaling is: