Transformation Techniques: Matrices, Projections, and Coordinate Systems

## Core Concepts

### 1. **Coordinate Systems and Homogeneous Coordinates**
#### Definitions
- **World Coordinate System (WCS)**: The global reference frame where objects are initially defined.
- **Viewing Coordinate System (VCS)**: The camera-centered frame after transformations.
- **Device Coordinate System (DCS)**: The frame for output (e.g., screen pixels).
- **Homogeneous Coordinates**: Extend 2D/3D points to 3D/4D using a 1 as the last coordinate. Enables matrix multiplication for transformations.

#### Why Homogeneous Coordinates?
```mermaid
graph TD
    A["2D Point (x,y)"] -->|Extend| B["Homogeneous: (x,y,1)"]
    C["3D Point (x,y,z)"] -->|Extend| D["Homogeneous: (x,y,z,1)"]
    B -->|Matrix Multiply| E["Transformed Point"]
    D -->|Matrix Multiply| E
    E -->|Project| F["Screen Coordinates"]

Example: 2D Translation

Convert translation $T_x,T_y$ to a matrix: $$\left[\begin{array}{ccc}1& 0& T_x\\ 0& 1& T_y\\ 0& 0& 1\end{array}\right]$$ Apply to point $(x,y)$: $$\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 3\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]=\left[\begin{array}{c}x+2\\ y+3\\ 1\end{array}\right]$$

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2. Basic 2D Transformations

Translation

• Matrix: $$T=\left[\begin{array}{ccc}1& 0& t_x\\ 0& 1& t_y\\ 0& 0& 1\end{array}\right]$$
• Example: Move point $(3,4)$ by $(5,-2)$ → $(8,2)$.

Rotation (θ around origin)

• Matrix: $$R=\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right]$$
• Example: Rotate $(1,0)$ by $90^{\circ }$ → $(0,1)$.

Scaling (s_x, s_y)

• Matrix: $$S=\left[\begin{array}{ccc}{s}_x& 0& 0\\ 0& s_y& 0\\ 0& 0& 1\end{array}\right]$$
• Example: Scale $(2,3)$ by $(0.5,2)$ → $(1,6)$.

Reflection

• About x-axis: $$\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right]$$
• About line $y=mx+c$: Derive using rotation + reflection + inverse rotation (see past exam question).

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3. Composite Transformations

Combine transformations by multiplying matrices in reverse order (right-to-left application). Example: Rotate 90° then translate $(2,3)$: $$T\cdot R=\left[\begin{array}{ccc}0& -1& 2\\ 1& 0& 3\\ 0& 0& 1\end{array}\right]$$ Apply to $(x,y)$: $$\left[\begin{array}{ccc}0& -1& 2\\ 1& 0& 3\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]=\left[\begin{array}{c}-y+2\\ x+3\\ 1\end{array}\right]$$

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4. Window-to-Viewport Transformation

Definitions

• Window: Sub-region of WCS to display (e.g., $A(20,20)$ to $D(90,70)$).
• Viewport: Target region on screen (e.g., $0\le x\le 640$, $0\le y\le 480$).
• Transformation: $$x_{vp}=\frac{x_w-x_{min}}{x_{max}-x_{min}}\cdot (v{p}_{max}-v{p}_{min})+v{p}_{min}$$ $$y_{vp}=\frac{y_w-y_{min}}{y_{max}-y_{min}}\cdot (v{p}_{max}-v{p}_{min})+v{p}_{min}$$

Example

Map window $[20,90]\times [20,70]$ to viewport $[100,500]\times [100,300]$:

• For $x_w=55$: $$x_{vp}=\frac{55-20}{70}\cdot 400+100=300$$

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5. Coordinate System Conversions

World → Viewing Coordinate System

1. Translation: Move origin to camera position.
2. Rotation: Align axes with camera orientation.
3. Scaling: Adjust for perspective (optional).

Matrix Form: $$M_{view}=T\cdot R\cdot S$$

Viewing → Device Coordinates

• Perspective Projection: Simulate 3D depth (divide by $z$).
• Orthographic Projection: Parallel projection (no depth distortion).

Comparison Table:

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6. 3D Transformations

Extend 2D matrices to 4x4:

• Translation: $$\left[\begin{array}{cccc}1& 0& 0& t_x\\ 0& 1& 0& t_y\\ 0& 0& 1& t_z\\ 0& 0& 0& 1\end{array}\right]$$
• Rotation about X-axis (θ): $$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \theta & -\sin \theta & 0\\ 0& \sin \theta & \cos \theta & 0\\ 0& 0& 0& 1\end{array}\right]$$

Example: Rotate point $(1,1,1)$ by $90^{\circ }$ about X-axis → $(1,0,1)$.

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Exam Tip

1. Matrix Multiplication Order: Remember transformations are applied right-to-left (e.g., T * R * S means scale first, then rotate, then translate).
2. Homogeneous Coordinates: Always use 4D vectors for 3D transformations to avoid errors.
3. Window-Viewport: Derive the formula from linear interpolation—examiners love this step.
4. Composite Transformations: For past exam questions (e.g., reflection about $y=mx+c$), break it into:
   • Rotate to align line with axis.
   • Reflect.
   • Rotate back.
5. Diagrams: Sketch the before/after of transformations (e.g., rotation of a square). Label axes and angles.
6. Common Pitfalls:
   • Forgetting to normalize homogeneous coordinates (divide by $w$).
   • Mixing up orthographic/perspective projection matrices.
   • Incorrectly handling negative scales (e.g., reflection).

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A 4x4 homogeneous transformation matrix with labeled components for translation, rotation, and scaling. (Image: D.stebani, CC BY-SA 4.0, via Wikimedia Commons)