2D Geometrical Transformations

1. Introduction

In computer graphics, transformations are used to manipulate objects on the screen. 2D transformations operate on objects defined in a two-dimensional space.

• Modify position and orientation
• Essential for animation and design

2. Types of 2D Transformations

• Translation
• Scaling
• Rotation
• Reflection
• Shearing

3. Translation

Translation moves an object from one position to another by adding translation values to its coordinates.

x' = x + tx
y' = y + ty

• Shifts object horizontally or vertically
• No change in size or orientation

4. Scaling

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

x' = x × sx
y' = y × sy

• Uniform and non-uniform scaling
• Object may grow or shrink

5. Rotation

Rotation turns an object around a fixed point (usually the origin) by a specified angle.

x' = x cosθ − y sinθ
y' = x sinθ + y cosθ

• Clockwise or anticlockwise rotation
• Angle in degrees or radians

6. Reflection

Reflection produces a mirror image of an object about a line such as x-axis, y-axis, or origin.

• Reflection about x-axis
• Reflection about y-axis
• Reflection about origin

7. Shearing

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

x' = x + shy
y' = y + shx

• Horizontal shearing
• Vertical shearing

8. Matrix Representation

2D transformations are represented using matrices, which allow multiple transformations to be combined efficiently.

• Uses homogeneous coordinates
• 3×3 transformation matrices

9. Applications

• 2D animation
• Computer-aided design (CAD)
• Game graphics

Practice Questions

1. What are 2D transformations?
2. Explain translation and scaling.
3. Write rotation equations.
4. Define reflection and shearing.
5. Where are 2D transformations used?

Practice Task