Two-Dimensional Transformations in Computer Graphics – Explained with Examples

Transformation:

Fundamental to all computer graphics system is the ability to simulate the manipulation of objects in space. this simulated spatial manipulation is referred to as transformation

Geometric transformations:

Geometric transformations in computer graphics are mathematical operations used to manipulate the position, size, orientation, and shape of objects within a 2D or 3D space.

Coordinate fixed, object move

Coordinate transformation:

Coordinate transformation is used to move objects from one coordinate system to another. This is done by transforming the coordinates of the object from one system to another, allowing the object to be moved, rotated, and scaled in a virtual environment.

Coordinate change, object fixed

Translation:

Moves an object from one location to another without changing its shape or size.

P’ = T_v(p)

P = , P’ = , T_v  =

 =  

=

x’ = x + t_x, y’ = y + t_y

 

Rotation:

Turns an object around a fixed point at a certain angle.

P’ = R_Ꝋ(p)

P = , P’ = , R_Ꝋ  =

 =  

=

x’ = x cos(Ꝋ) – y sin(Ꝋ), y’ = x sin(Ꝋ) + y cos(Ꝋ)

Scaling:

Enlarges or reduces the size of an object proportionally.

P’ = (S_xS_y) p

P = , P’ = , S_xS_y =

 =  

=

x’ = xs_x, y’ = ys_y

 

 

Mirroring (Reflection):

Flips an object over a line to produce a mirror image.

m_x =

the mirror reflection transformation   m_x, about the x axis is given by

P’ = m_xp

where x’ = x and y’ = -y

similarly, the mirror reflection about the y axis is

p’ = m_y(p)

where x’ = -x and y’ = y

 

 

Math:

v    Transform the triangle with vertices A(0,0), B(1,1) and C (5,2) with V = 5I + 2J respect to origin

We know,

               P’ = T_v(p)

 P’ =

, T_v  =

P=

 

P’ =

   =

thus A' =(5,2), B' = (6,3), C' = (10, 4)