3D Geometrical Transformations

1. Introduction

In 3D graphics, objects are defined using x, y, and z coordinates. Transformations modify these coordinates to move, resize, or rotate objects in 3D space.

• Essential for 3D rendering
• Used in modeling and animation

2. Types of 3D Transformations

• Translation
• Scaling
• Rotation
• Reflection
• Shearing

3. Translation

Translation moves an object in 3D space by adding translation values to its coordinates.

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

• Moves object along x, y, z axes
• No change in size or orientation

4. Scaling

Scaling changes the size of an object by multiplying its coordinates with scaling factors.

x' = x × sx
y' = y × sy
z' = z × sz

• Uniform and non-uniform scaling
• Can enlarge or shrink objects

5. Rotation

Rotation turns an object about one of the coordinate axes (X, Y, or Z).

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

6. Rotation About Coordinate Axes

Z-axis Rotation:
x' = x cosθ − y sinθ
y' = x sinθ + y cosθ
z' = z

• Similar matrices for X and Y axes
• Angle θ in degrees or radians

7. Reflection

Reflection produces a mirror image of an object about a plane such as XY, YZ, or XZ plane.

• Reflection about XY-plane
• Reflection about YZ-plane
• Reflection about XZ-plane

8. Shearing

Shearing distorts the shape of an object by shifting one coordinate proportional to another.

• X-shear, Y-shear, Z-shear
• Used for distortion effects

9. Matrix Representation

3D transformations are represented using 4×4 homogeneous transformation matrices.

• Uses homogeneous coordinates
• Supports composite transformations

10. Applications

• 3D animation and modeling
• Game graphics
• Virtual reality and simulations

Practice Questions

1. What are 3D geometrical transformations?
2. Explain 3D translation and scaling.
3. Describe rotation about coordinate axes.
4. What is reflection in 3D?
5. Where are 3D transformations used?

Practice Task