Triangle Rasterization

📘 Triangle Rasterization — Exam Notes (Computer Graphics)

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🔹 1. Introduction

Triangle Rasterization is the process of converting a triangle (defined by 3 vertices in 2D/3D space) into a set of pixels on the screen.

👉 In simple words: It decides which pixels inside a triangle should be colored to display it on a raster screen.

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🔹 2. Why Triangles?

Triangles are used because:

• Always planar (flat surface)
• Simple mathematical processing
• Efficient for GPUs
• Basis of all 3D models (triangle mesh)

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🔹 3. Basic Idea of Rasterization

Given triangle vertices: [ A(x_1,y_1),; B(x_2,y_2),; C(x_3,y_3) ]

👉 Goal:

• Fill all pixels inside the triangle boundary

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🔹 4. Steps in Triangle Rasterization

✔️ Step 1: Find Bounding Box

• Determine smallest rectangle covering triangle: [ x_{min}, x_{max}, y_{min}, y_{max} ]

👉 Only pixels inside this box are checked.

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✔️ Step 2: Scan Each Pixel

• Check every pixel inside bounding box

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✔️ Step 3: Inside-Outside Test

• Decide whether pixel lies inside triangle

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✔️ Step 4: Color Pixel

• If inside → fill pixel
• If outside → ignore

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🔹 5. Methods of Triangle Rasterization

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🔸 5.1 Edge Function Method (Very Important)

✔️ Concept

A point is inside triangle if it is on the same side of all edges.

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✔️ Edge Equation

For edge between two points: [ E(x,y) = Ax + By + C ]

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✔️ Rule

For triangle ABC:

• Compute edge functions:

  • AB, BC, CA

• If all have same sign → point is inside

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✔️ Advantages

• Fast
• Used in GPUs
• Easy parallelization

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🔸 5.2 Barycentric Coordinate Method

✔️ Concept

Any point inside triangle can be written as:

[ P = \alpha A + \beta B + \gamma C ]

Where:

• (\alpha + \beta + \gamma = 1)

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✔️ Inside Condition

Point is inside if: [ 0 \le \alpha, \beta, \gamma \le 1 ]

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✔️ Advantages

• Used for interpolation (color, texture, depth)
• Very accurate

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🔸 5.3 Scan Line Algorithm

✔️ Concept

• Draw horizontal lines (scan lines)
• Find intersection points with triangle edges
• Fill pixels between intersections

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✔️ Steps

1. For each scan line (row):
2. Find intersection with triangle edges
3. Sort intersection points
4. Fill pixels between pairs

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✔️ Advantage

• Efficient for filled polygons

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🔹 6. Pixel Coverage Concept

• Each pixel has a center point
• If center lies inside triangle → pixel is filled

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🔹 7. Important Tests (Inside Triangle)

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✔️ 7.1 Area Method

Idea:

If point is inside triangle:

[ Area(ABC) = Area(PBC) + Area(APC) + Area(ABP) ]

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✔️ 7.2 Sign Method

• Check orientation using cross product
• Same sign → inside

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🔹 8. Interpolation (Important in Graphics)

Used to calculate:

• Color
• Texture
• Depth

Using barycentric coordinates.

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🔹 9. Applications

• 3D rendering
• Games
• Animation
• GPU pipelines
• CAD systems

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🔹 10. Advantages

• Efficient rendering
• Works well with hardware (GPU)
• Forms basis of real-time graphics

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🔹 11. Limitations

• Large number of pixels to process
• Requires optimization for real-time rendering

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🔹 12. Diagram Descriptions

✔️ Triangle Rasterization Grid

• Draw triangle on pixel grid
• Show filled pixels inside

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✔️ Bounding Box

• Rectangle surrounding triangle

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✔️ Scan Lines

• Horizontal lines filling triangle

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✔️ Edge Test

• Show pixel checked against triangle edges

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📝 Likely Exam Questions

1. Define triangle rasterization.
2. Why are triangles used in graphics?
3. Explain bounding box method.
4. What is edge function method?
5. Explain barycentric coordinates.
6. Describe scan line algorithm.
7. How do you determine if a point lies inside a triangle?
8. What is pixel coverage?
9. Explain interpolation in rasterization.
10. Compare different rasterization methods.

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⚡ Quick Revision Summary

• Triangle rasterization = converting triangle into pixels

• Steps:

  • Find bounding box
  • Test pixels
  • Fill inside pixels

• Methods:

  • Edge function (fastest, GPU-based)
  • Barycentric coordinates (used for interpolation)
  • Scan line algorithm (horizontal filling)

• Core idea: decide inside vs outside pixels

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