📘 Perspective Projection — Exam Notes
🔹 1. Introduction
Perspective Projection is a technique used in computer graphics to convert 3D objects into 2D images in a way that mimics human vision.
👉 Objects that are farther away appear smaller, creating a sense of depth and realism.
🔹 2. Basic Concept
- All projection lines (projectors) meet at a single point called the:
✔️ Center of Projection (COP)
- Also known as Eye Point or Viewer Position
- The object is projected onto a plane called the:
✔️ Projection Plane (View Plane)
👉 Flow:
3D Object → Projectors → Projection Plane → 2D Image
🔹 3. Key Characteristics
- Realistic appearance
- Shows depth perception
- Parallel lines may converge
- Size decreases with distance
🔹 4. Types of Perspective Projection
🔸 4.1 One-Point Perspective
✔️ Definition
✔️ Features
- Only one set of parallel lines converges
- Used when object faces viewer directly
✔️ Example
- Railway tracks appearing to meet at a point
🔸 4.2 Two-Point Perspective
✔️ Definition
✔️ Features
- Two sets of parallel lines converge
- Used for corner views of objects
🔸 4.3 Three-Point Perspective
✔️ Definition
✔️ Features
- Includes vertical convergence
- Used for tall buildings or dramatic views
🔹 5. Mathematical Representation
✔️ Simple Perspective Projection Formula
For a point ((x, y, z)):
[
x' = \frac{x}{z}, \quad y' = \frac{y}{z}
]
👉 This shows:
- Larger (z) (farther away) → smaller image
✔️ General Form (with distance (d))
[
x' = \frac{d \cdot x}{z}, \quad y' = \frac{d \cdot y}{z}
]
Where:
- (d) = distance from viewer to projection plane
🔹 6. Perspective Transformation Matrix
[
\begin{bmatrix}
1 & 0 & 0 & 0
0 & 1 & 0 & 0
0 & 0 & 1 & 0
0 & 0 & \frac{1}{d} & 0
\end{bmatrix}
]
👉 After transformation, perform division by z (or w).
🔹 7. Vanishing Point
✔️ Definition
Point where parallel lines appear to meet.
✔️ Importance
- Creates depth illusion
- Key element in perspective drawing
🔹 8. Comparison with Parallel Projection
| Feature |
Perspective Projection |
Parallel Projection |
| Realism |
High |
Low |
| Parallel lines |
Converge |
Remain parallel |
| Depth perception |
Yes |
No |
| Used in |
Games, movies |
Engineering |
🔹 9. Diagram Descriptions
✔️ Perspective Projection Setup
-
Draw:
- Eye (COP)
- Object
- Projection plane
- Lines from eye to object
✔️ Vanishing Point
- Draw parallel lines meeting at a point
✔️ One/Two/Three Point Perspective
- Show convergence of lines
🔹 10. Applications
- 3D games
- Animation movies
- Virtual reality
- Architectural visualization
- Simulation systems
🔹 11. Advantages
- Realistic images
- Depth perception
- Natural viewing
🔹 12. Limitations
- Complex calculations
- Distortion of object size
- Not suitable for technical drawings
🔹 13. Important Terms
- COP (Center of Projection)
- Projection Plane
- Vanishing Point
- Projectors
- Foreshortening (objects appear shorter at distance)
📝 Likely Exam Questions
- Define perspective projection.
- Explain types of perspective projection.
- What is a vanishing point?
- Derive perspective projection formula.
- Compare perspective and parallel projection.
- Explain the role of COP.
- What is foreshortening?
- Draw and explain one-point perspective.
⚡ Quick Revision Summary
-
Perspective projection creates realistic 3D view
-
Based on Center of Projection (COP)
-
Formula:
[
x' = \frac{d \cdot x}{z},\quad y' = \frac{d \cdot y}{z}
]
-
Types:
- One-point
- Two-point
- Three-point
-
Parallel lines meet at vanishing point
-
Used in games, animation, VR