Geometric transformations include translation, rotation, and scaling. Translation moves every point of an object by adding a translation vector. Rotation rotates the object around an axis by a certain angle. Scaling enlarges or shrinks the object by a scale factor. More complex transformations can be achieved by combining these basic transformations through composition.

1. 3-D Geometric Transformations
Geometric Transformation : The object itself
is moved relative to a stationary coordinate
system or background.
With respect to some 3-D coordinate system, an
object Obj is considered as a set of points.
Obj = { P(x,y,z)}
If the Obj moves to a new position, the new
object Obj’ is considered:
Obj’ = { P’(x’,y’,z’)}

2. Translation
Moving an object is called a translation. We
translate an object by translating each
vertex in the object.
x’ = x + tx
y’ = y + ty
z’ = z + tz

3. The translating distance pair( tx, ty, tz) is
called a translation vector or shift vector.
We can also write this equation in a single
Matrix using column vectors:
x’ 1 0 0 tx x
y’ = 0 1 0 ty y
z’ 0 0 1 tz z
1 0 0 0 1 1

4. Rotation
In 2-D, a rotation is prescribed by an angle θ
& a center of rotation P. But in 3-D
rotations require the prescription of an
angle of rotation & an axis of rotation.
 Rotation about the z axis:
R θ,K  x’ = x cosθ – y sinθ
y’ = x sinθ – y cosθ
z’ = z

5. Rotation about the y axis:
R θ,J  x’ = x cosθ + z sinθ
y’ = y
z’ = - x sinθ + z cosθ
Rotation about the x axis:
R θ,I  x’ = x
y’ = y cosθ – z sinθ
z’ = y sinθ + z cosθ

6. & the rotation matrix corresponding is
cos θ -sin θ 0
R θ,K = sin θ cos θ 0
0 0 1
cos θ 0 sin θ
R θ,J = 0 1 0
-sin θ 0 cos θ

7. 1 0 0
R θ,I = 0 cos θ -sin θ
0 sin θ cos θ

8. Scaling
Changing the size of an object is called
Scaling . The scale factor s determines
whether the scaling is a magnification, s > 1,
Or a reduction, s < 1. Scaling with respect to
the origin, where the origin remains fixed,
x’ = x . sx
Ssx,sy,sz  y’ = y . sy
z’ = z . sz

9. The transformation equations can be written
in the matrix form:
x’ sx 0 0 x
y’ = 0 sy 0 . y
z’ 0 0 sz z

10. Coordinate Transformation
Translation
If the xyz coordinate system is displaced to a
new position, the coordinates of a point in
both systems are related by the translation
Transformation:
Tv  (x’,y’,z’) = Tv (x,y,z)
where x’ = x – tx,
y’ = y – ty , z’ = z – tz

11. In matrix notation,
1 0 0 -tx
Tv = 0 1 0 -ty
0 0 1 -tz
0 0 0 1
Similarly, we can express the coordinate
scaling & rotation transformations.

12. Composite Transformation
More complex geometric and coordinate
transformations are formed the process of
composition of functions.
Rotation About an Arbitrary Axis in space:
1.) Translate the object so that the rotation
axis passes through the coordinate origin.
2.) Rotate the object so that the axis of
rotation coincides with one of the
coordinate axes.

13. 3.) Perform the specific rotation about the
coordinate axis.
4.) Apply inverse rotations to bring the
rotation axis back to its original
orientation.
5.) Apply the inverse translation to bring the
rotation axis back to its original position.
We can transform the rotation axis onto any
of the three coordinate axes. For eg. We
are taking rotation onto the z-axis.

14. y
P2 P2
P1’
x
P1 P1’
z P2’’
P2
P2
P1’
P1’ P1
P2’’