This document discusses 3D transformations in computer graphics. It describes how 3D transformations modify and reposition graphics in 3D space using translation, scaling, and rotation. Translation moves an object using vectors in the x, y, and z directions. Scaling changes an object's size using scaling factors for each axis. Rotation changes an object's angle by specifying a rotation axis, direction, and angle. Matrix representations of transformations are provided.

1. Pooja Dixit
Department of Computer Science
Sophia Girls’ College (Autonomous),
Ajmer
3-D Transformation

2. 3-D Transformation
 In Computer graphics , Transformation is a process of
modifying, re-positioning and change the size of the existing
graphics.
 In the 2D system, we use only two coordinates X and Y but
in 3D, an extra coordinate Z is added.
 3D graphics techniques and their application are
fundamental to the Entertainment(movie making), Games,
and Computer-aided design industries.
Fig:3-D Coordinate

3. Types of 3-D Transformation
3-D Transformation
Techniques
Scaling
Translation
Rotation
Reflection
Shearing
Basic
Transformation
Techniques
Other
Transformation
Techniques

4. 3-D Translation
 Moving an object from one position to another position is called a
translation.
 Translation is done using translation vectors. There are three
vectors in 3D instead of two. These vectors are in x, y, and z
directions.
 Translation in the x-direction is represented using Tx.
 The translation is y-direction is represented using Ty.
 The translation in the z- direction is represented using Tz.
Figure:1

5. 3-D Translation
 If P is a point having X,Y,Z coordinates. So after translation
its coordinates will be (x1 y1 z1). And the translation vector
will be Tx Ty Tz for x,y, and z directions respectively.
x1=x+ Tx
y1=y+Ty
z1=z+ Tz
Matrix representation of point translation
Point shown in fig is (x, y, z). It become (x1,y1,z1) after translation.
Tx Ty Tz are translation vector.

6. 3-D Scaling
 Scaling is used to change the size of an object. The size can be
increased or decreased.
 The three scaling factors are required Sx Sy and Sz.
 Sx=Scaling factor in x- direction
Sy=Scaling factor in y-direction
Sz=Scaling factor in z-direction

7. 3-D Scaling
 Note: If all scaling factors Sx=Sy=Sz.Then scaling is called
as uniform.
 If scaling is done with different scaling vectors, it is called
a differential scaling.
 Scaling of the object relative to a fixed point
 Following are steps performed when scaling of objects with
fixed point (a, b, c). It can be represented as below:
1. Translate fixed point to the origin
2. Scale the object relative to the origin
3. Translate object back to its original position.

8. 3-D Scaling
 In figure (a) point (a, b, c) is shown, and object whose scaling is to
done also shown in steps in fig (b), fig (c) and fig (d).


9. 3-D Scaling
 Scaling Equations:
 x1=x · sx
 y1=y · sy
 z1=z · sz
 Scaling Matrix:

10. 3-D Rotation
 It is a process of changing the angle of the object. Rotation can be
clockwise or anticlockwise. For rotation, we have to specify the
angle of rotation and rotation point. Rotation point is also called a
pivot point. It is print about which object is rotated.
 3-D Rotation is a little more complex than 2-D Rotation because
you have to specify multiple items of information. The following
three items are required:
 Rotation axis (the axis the shape will be rotated around)
 Rotation direction (the direction: clockwise or counterclockwise)
 Rotation angle (the number of degrees the shape will be rotated through

11. 3-D Rotation
 Rotation about X-axis Anticlockwise:
 This rotation is achieved by using
the following rotation equations-
 Xnew = Xold
 Ynew = Yold x cosθ – Zold x sinθ
 Znew = Yold x sinθ + Zold x cosθ
 In Matrix form, the above rotation equations may be represented as-

12. 3-D Rotation
 For Y-Axis Rotation-
 This rotation is achieved by using the following
rotation equations-
 Xnew = Zold x sinθ + Xold x cosθ
 Ynew = Yold
 Znew = Yold x cosθ – Xold x sinθ
 In Matrix form, the above rotation equations may be represented as-

13. 3-D Rotation
 For Z-Axis Rotation-
 This rotation is achieved by using the
 following rotation equations-
Xnew = Xold x cosθ – Yold x sinθ
Ynew = Xold x sinθ + Yold x cosθ
Znew = Zold
 In Matrix form, the above rotation equations may be represented
as-

14. Thank You