Transformations & Windowing and Clipping

Basic Transformations

Transformations are operations that change the position, size, or orientation of objects.

Types of Basic Transformations:

1. Translation (Shifting)

• Moves object from one position to another

$$x' = x + t_x,\quad y' = y + t_$$

Example: Move a shape 5 units right, 3 units up

2. Scaling (Resizing)

• Changes size of object

$$x' = x \cdot S_x,\quad y' = y \cdot S$$

👉 $S > 1$: enlarge, $S < 1$: shrink

3. Rotation (Turning)

• Rotates object around origin

$$x' = x\cos\theta - y\sin\theta$$$$y' = x\sin\theta + y\cos\thet$$

👉 Angle measured anticlockwise

Matrix Representation & Homogeneous Coordinates

Why Matrix?

• Makes transformations easy and consistent
• Allows combining multiple transformations

Matrix Forms

Translation

$$\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}$$

Scaling

$$\begin{bmatrix} S_x & 0 & 0 \\ 0 & S_y & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Rotation 

$$\begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

​Homogeneous Coordinates:

• Add extra coordinate (w = 1)
• Represent point as (x, y, 1)

Benefits

• Enables translation using matrix multiplication
• Simplifies combining transformations

Composite Transformations

Combining multiple transformations into a single transformation matrix.

Key Concept

Order matters!

👉 Example:

• Rotate → Translate ≠ Translate → Rotate

Process

1. Multiply transformation matrices
2. Apply final matrix to object

Advantages:

• Faster computation
• Efficient rendering

Reflection

Creates a mirror image of an object.

Types

1. Reflection about X-axis:

$$(x, y) \rightarrow (x, -y$$

2. Reflection about Y-axis: 

$$(x, y) \rightarrow (-x, y$$

3. Reflection about Origin: 

$$(x, y) \rightarrow (-x, -y$$

4. Reflection about line $y = x$: 

$$(x, y) \rightarrow (y, x$$

Shearing

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

Types:

1. X-Shear:

$$x' = x + Sh_x \cdot$$

👉 Object shifts horizontally

2. Y-Shear: 

$$y' = y + Sh_y \cdot x$$

👉 Object shifts vertically

Effect

• Rectangle → Parallelogram

Final Summary Table

Transformation	Purpose	Key Effect
Translation	Move object	Position change
Scaling	Resize	Size change
Rotation	Turn object	Orientation change
Reflection	Mirror image	Flip
Shearing	Distort shape	Skew

Concept Flow

Basic Transformations → Matrix Form → Homogeneous Coordinates → Composite Transformations → Advanced (Reflection & Shearing)

Viewing Pipeline

The viewing pipeline is the sequence of steps that converts objects from world coordinates to screen coordinates.

Stages:

1. Modeling Transformation
   • Object → world coordinates
2. Viewing Transformation
   • Select portion of scene (camera view)
3. Clipping
   • Remove unwanted parts
4. Normalization
   • Map to standard coordinate system
5. Viewport Transformation
   • Map to screen/display

Purpose:

• Display only the visible portion
• Improve efficiency

Viewing Transformations

Transformations used to map world coordinates to viewing coordinates.

Key Concepts:

1. Window

• Area of world we want to see

2. Viewport

• Area on screen where it is displayed

Window → Viewport Mapping: 

$$x_v = x_{vmin} + \frac{(x - x_{wmin})(x_{vmax}-x_{vmin})}{x_{wmax}-x_{wmin}$$

(Similarly for y)

Importance

• Controls zooming and panning
• Maintains aspect ratio

2D Clipping Algorithms

Clipping removes parts of objects that lie outside the viewing window.

Line Clipping Algorithms

A. Cohen-Sutherland Line Clipping

Idea

• Divide space into 9 regions
• Assign 4-bit region codes

Steps

1. Assign region codes to endpoints
2. If both inside → accept
3. If both outside (same region) → reject
4. Else → find intersection

Advantages

• Simple
• Efficient for trivial cases

Disadvantages

• More calculations for complex cases

B. Liang-Barsky Algorithm

Idea

• Uses parametric line equations
• Avoids repeated intersection calculations

Equation: 

$$x = x_1 + t(x_2 - x_1)$$$$y = y_1 + t(y_2 - y_1$$

Advantages

• Faster than Cohen-Sutherland
• More efficient

Disadvantages:

• Slightly complex

C. Line Clipping Against Non-Rectangular Windows

Idea

• Clip lines against arbitrary shapes (polygon, circle)

Methods:

• Intersection with each boundary
• Use polygon clipping techniques

Polygon Clipping

A. Sutherland-Hodgman Polygon Clipping

Idea

• Clip polygon edge by edge against window

Steps:

1. Take one boundary
2. Process all edges
3. Generate new polygon
4. Repeat for all boundaries

Advantages:

• Simple
• Works well for convex polygons

Disadvantages

• Not suitable for concave polygons

B. Weiler-Atherton Polygon Clipping

Idea:

• Handles complex and concave polygons

Key Concept

• Uses entry and exit points
• Traverses polygon and window alternately

Advantages

• Works for all polygon types

Disadvantages

• Complex implementation

Curve Clipping

Clipping curves (circle, ellipse, spline) against a window.

Methods

• Approximate curve with line segments
• Clip each segment
• Or use mathematical boundary checks

Challenge

• More complex than line clipping

Text Clipping

Clipping text based on its position relative to window.

Types

1. All-or-None String Clipping

• Entire text shown or removed

2. All-or-None Character Clipping

• Individual characters shown/hidden

3. Component Clipping

• Clip individual character shapes

Final Summary Table

Topic	Key Idea	Best For
Cohen-Sutherland	Region codes	Simple cases
Liang-Barsky	Parametric	Efficient
Sutherland-Hodgman	Edge clipping	Convex polygons
Weiler-Atherton	Entry/exit	Complex polygons
Curve Clipping	Segment approximation	Curves
Text Clipping	Character/string level	Text rendering

Concept Flow

Viewing Pipeline → Window & Viewport → Clipping → Line → Polygon → Curve → Text