3D Scaling, Rotation & Translation

1. Introduction

In 3D graphics, objects must be resized, rotated, and moved to create realistic scenes. These operations are performed using transformation matrices.

• Fundamental 3D operations
• Used in modeling and animation

2. 3D Scaling

3D scaling changes the size of an object by multiplying its coordinates with scaling factors along x, y, and z axes.

x' = x × sx
y' = y × sy
z' = z × sz

• Uniform scaling: sx = sy = sz
• Non-uniform scaling: different factors

3. Scaling Matrix

[ sx  0   0   0 ]
[ 0   sy  0   0 ]
[ 0   0   sz  0 ]
[ 0   0   0   1 ]

4. 3D Translation

Translation moves an object from one location to another in 3D space without changing its size or orientation.

x' = x + tx
y' = y + ty
z' = z + tz

• Movement along coordinate axes
• Uses homogeneous coordinates

5. Translation Matrix

[ 1  0  0  tx ]
[ 0  1  0  ty ]
[ 0  0  1  tz ]
[ 0  0  0  1  ]

6. 3D Rotation

Rotation changes the orientation of an object about a specific axis in 3D space.

• Rotation about X-axis
• Rotation about Y-axis
• Rotation about Z-axis

7. Rotation About Z-Axis

[ cosθ  -sinθ  0  0 ]
[ sinθ   cosθ  0  0 ]
[  0       0   1  0 ]
[  0       0   0  1 ]

• Rotation in XY-plane
• Angle θ in radians

8. Order of Transformations

The order of applying scaling, rotation, and translation affects the final position and orientation of objects.

• Matrix multiplication is not commutative
• Order must be carefully chosen

9. Advantages

• Precise object control
• Flexible scene manipulation
• Essential for 3D graphics pipeline

10. Applications

• 3D modeling and animation
• Game development
• Virtual reality systems

Practice Questions

1. What is 3D scaling?
2. Write 3D translation equations.
3. Explain rotation about Z-axis.
4. Why is order of transformations important?
5. Where are 3D transformations used?

Practice Task