Perspective Projection

📘 Perspective Projection — Exam Notes

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🔹 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.

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🔹 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

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• The object is projected onto a plane called the:

✔️ Projection Plane (View Plane)

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👉 Flow:

3D Object → Projectors → Projection Plane → 2D Image

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🔹 3. Key Characteristics

• Realistic appearance
• Shows depth perception
• Parallel lines may converge
• Size decreases with distance

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🔹 4. Types of Perspective Projection

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🔸 4.1 One-Point Perspective

✔️ Definition

• One vanishing point

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✔️ Features

• Only one set of parallel lines converges
• Used when object faces viewer directly

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✔️ Example

• Railway tracks appearing to meet at a point

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🔸 4.2 Two-Point Perspective

✔️ Definition

• Two vanishing points

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✔️ Features

• Two sets of parallel lines converge
• Used for corner views of objects

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🔸 4.3 Three-Point Perspective

✔️ Definition

• Three vanishing points

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✔️ Features

• Includes vertical convergence
• Used for tall buildings or dramatic views

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🔹 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

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✔️ 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

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🔹 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).

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🔹 7. Vanishing Point

✔️ Definition

Point where parallel lines appear to meet.

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✔️ Importance

• Creates depth illusion
• Key element in perspective drawing

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🔹 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

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🔹 9. Diagram Descriptions

✔️ Perspective Projection Setup

• Draw:

  • Eye (COP)
  • Object
  • Projection plane
  • Lines from eye to object

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✔️ Vanishing Point

• Draw parallel lines meeting at a point

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✔️ One/Two/Three Point Perspective

• Show convergence of lines

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🔹 10. Applications

• 3D games
• Animation movies
• Virtual reality
• Architectural visualization
• Simulation systems

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🔹 11. Advantages

• Realistic images
• Depth perception
• Natural viewing

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🔹 12. Limitations

• Complex calculations
• Distortion of object size
• Not suitable for technical drawings

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🔹 13. Important Terms

• COP (Center of Projection)
• Projection Plane
• Vanishing Point
• Projectors
• Foreshortening (objects appear shorter at distance)

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📝 Likely Exam Questions

1. Define perspective projection.
2. Explain types of perspective projection.
3. What is a vanishing point?
4. Derive perspective projection formula.
5. Compare perspective and parallel projection.
6. Explain the role of COP.
7. What is foreshortening?
8. Draw and explain one-point perspective.

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⚡ 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

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