This document discusses various geometric transformations used in computer graphics including translation, rotation, scaling, shearing, reflection, and their mathematical representations. It also covers topics like matrix representation, homogeneous coordinates, and composite transformations. Translation moves an object along a straight path. Rotation repositions an object around a circular path by a specified angle. Scaling changes an object's size. Shearing distorts the shape of an object. Reflection produces a mirror image. Matrix representation stores transformations. Homogeneous coordinates provide a unified treatment of transformations and projections. Composite transformations combine multiple transformations into one.

1. Computer Graphics
By
Ms.N.RUBA
Asst. Prof / CA
BSCW, Thanjavur.

2. BASIC TRANSFORMATION
Transactions are helpful in changing the
position,size,orientation,shape,etc.,of the
object. There are three basic rigid
transformation.
*Translation
*Rotation
*Scaling
The derived geometrical transformation is:
*Reflection
*Shearing

3. The translation is repositioning an object along a straight-line path
from one coordinate location to another coordinate location.
The translation is the rigid body transformation that saves an object
without deformation. A translation moves to a different position on the
screen.
X’=X + tx
Y’=Y + ty
TRANSLATION
P=[X]/[Y] P’=[X’][Y’]
we can also write as:
P’=P+T
T=[tx]/[ty]

4. ROTATION
It is the transformation that is used to
reposition one object along the circular path in
the XY plane. We specify a rotation angle TITA
and the portion of the rotation point A and B
about which the object is being rotated to
generate a rotation.
P’=P.R
Where R is the rotation matrix

5. SCALING
Scaling is the transformation that is used to change the
object’s size. The Operation is carried out by multiplying
the coordinate value(X,Y)with Sx and Sy scaling factors.
X’=X. Sx and Y’=Y. Sy
P’=P . S

6. SHEARING
Shearing is the transformation used to change the
shape of an existing object in the 2D plane. The size of
the object changes along the x direction as well as the
Y direction.
Reflection
Reflection is the mirror image of the original object.
In other words, we will say that it is the rotation
operation with 180 degree. In reflection
transformation, the object’s size does not change.

8. MATRIX REPRESENTATION
Matrix representation is a method used by
a computer language to store matrices of
more than one dimension in memory. Fortran
and C use different schemes for their native
arrays. Fortran uses “Column Major”, in
which all the elements for a given column are
stored contiguously in memory.
P’=M1 +P +M2
x=xh/h+ y=yh/h

9. HOMOGENEOUS COORDINATE
Homogeneous coordinates have a natural
application to computer graphics ; they form a
basis for the projective geometry used
extensively to project a three dimensional
scene onto a two dimensional image plane.
They also unify the treatment of common
graphical transformation and operations.
P’= T(t1…….tp)
P’=S(t1……….tp) P

10. HOMOGENEOUS COORDINATE EXAMPLES

11. COMPOSITE TRANSFORMATION
A number of transformation or sequence of
transformation can be combined into single
one called as composition. The process of
combining is called as concatenation.
Suppose we want to perform rotation about an
arbitrary point, then we can perform it by the
sequence of three transformation.
1. Translation
2. Rotation
3. Reverse Translation

12. EXAMPLE SHOWING COMPOSITE
TRANSFORMATION

13. THANK YOU