1. Lecture 10 Geometric Transformations In 3D(Three- Dimensional)
Computer Graphics
Lecture 10
Geometric Transformations In 3D(Three- Dimensional)

2. 3D Geometrical Transformations
3D Coordinate Systems
3D point representation
3D Translation
3D Scaling
3D reflection
3D Shearing
3D Rotations about x, y and z axis

3. 3D Coordinate Systems

4. 3D point representation
In 2D we represent a point P as
P = [x, y] or x y
And when it is converted into homogeneous coordinates, it is written as
P = [x, y, h] or x
y where h = 1 h
A 3D point P is represented as
P = [x, y, z] or x z

5. A 3D point P is represented in homogeneous coordinates by a 4-dimensional vector:
P = [x, y, z, h] or P = x
y where h=1 z h
P = x y 1

10. 3D Rotations In 2D, rotation matrix is given as x’ = x cosθ -sinθ
y’ y sinθ cosθ
In homogeneous system,
x’ = x cosθ sinθ
y’ y sinθ cosθ 0

11. In 3D, rotation matrix is given as
x’ = x cosθ sinθ
y’ y sinθ cosθ
z’ z

12. 3D Rotation about z-axis
The equations for 3D rotation about z-axis is given as
x’ = x cosθ – y sinθ
y’ = x sinθ + y cosθ
z’ = z

14. 3D Rotation about x-axis
The equations for 3D rotations about x-axis and y-axis can be obtained by replacements that is,
x  y  z  x
That is, equations for 3D rotation about x-axis are:
y’ = y cosθ – z sinθ
z’ = y sinθ + z cosθ
x’ = x

16. 3D Rotation about y-axis
Equations for 3D rotation about y-axis are:
z’ = z cosθ – x sinθ
x’ = z sinθ + x cosθ
y’ = y