Triangles and Planes

📘 Triangles and Planes — Exam Notes (Computer Graphics)

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🔹 1. Introduction

In computer graphics, triangles and planes are the most important geometric primitives used to represent 3D surfaces.

👉 Almost every 3D model (games, movies, CAD) is built using triangular meshes.

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🔹 2. Triangle in Computer Graphics

✔️ Definition

A triangle is a polygon formed by joining three non-collinear points in space:

[ P_1(x_1,y_1,z_1),; P_2(x_2,y_2,z_2),; P_3(x_3,y_3,z_3) ]

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✔️ Why Triangles are Important?

• Always planar (lie on a single plane)
• Simple to compute
• Stable for rendering
• Used to build complex 3D models

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✔️ Triangle Representation

A triangle has:

• 3 vertices
• 3 edges
• 1 surface

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🔹 3. Plane in Computer Graphics

✔️ Definition

A plane is a flat 2D surface extending infinitely in 3D space.

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✔️ General Equation of a Plane

[ Ax + By + Cz + D = 0 ]

Where:

• (A, B, C) = normal vector components
• (D) = constant

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🔹 4. Normal Vector of a Plane (Very Important)

✔️ Definition

A normal vector is a vector perpendicular to the plane.

[ \vec{N} = (A, B, C) ]

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✔️ Finding Normal from Triangle

Given points (P_1, P_2, P_3):

1. Find two vectors: [ \vec{U} = P_2 - P_1,\quad \vec{V} = P_3 - P_1 ]

2. Compute cross product: [ \vec{N} = \vec{U} \times \vec{V} ]

👉 This gives the plane’s normal vector.

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🔹 5. Equation of Plane Using Three Points

✔️ Steps

1. Take three points of triangle
2. Find two direction vectors
3. Compute cross product → normal vector
4. Substitute into plane equation

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🔹 6. Triangle Normal (Shading Use)

✔️ Importance

• Used in lighting and shading
• Determines how light reflects from surface

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✔️ Formula

[ \vec{N} = (P_2 - P_1) \times (P_3 - P_1) ]

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🔹 7. Plane Classification

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✔️ 7.1 Front Face / Back Face

• Determined using normal vector direction

If:

• Normal points toward viewer → Front face
• Away from viewer → Back face

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✔️ 7.2 Back-Face Culling (Important)

• Removes faces not visible to camera
• Improves rendering speed

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🔹 8. Point and Plane Relationship

For plane: [ Ax + By + Cz + D = 0 ]

Substitute point (P(x,y,z)):

✔️ Cases

• If value = 0 → Point lies on plane
• If value > 0 → One side of plane
• If value < 0 → Other side of plane

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🔹 9. Distance of Point from Plane

✔️ Formula

[ d = \frac{|Ax + By + Cz + D|}{\sqrt{A^2 + B^2 + C^2}} ]

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🔹 10. Plane in Rendering

Planes are used in:

• Polygon surfaces
• 3D object modeling
• Collision detection
• Lighting calculations

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🔹 11. Triangles in 3D Modeling

✔️ Triangle Mesh

• Complex objects are made of many triangles

• Example:

  • Human face → thousands of triangles
  • 3D car model → triangle mesh

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🔹 12. Advantages of Triangles

• Always planar
• Easy to compute
• Efficient rendering
• Works well with GPUs

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🔹 13. Limitations

• Large models require many triangles
• Can increase memory usage

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🔹 14. Diagram Descriptions

✔️ Triangle in 3D

• Draw 3 points in space
• Connect them to form triangle

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✔️ Plane

• Draw large flat surface
• Show normal vector perpendicular to surface

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✔️ Cross Product

• Show two vectors forming a plane
• Resulting normal vector perpendicular

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📝 Likely Exam Questions

1. Define triangle in computer graphics.
2. Why are triangles important in 3D modeling?
3. Write equation of a plane.
4. How do you find normal vector of a plane?
5. Explain plane equation using three points.
6. What is back-face culling?
7. Derive distance from point to plane.
8. Explain triangle mesh.
9. Differentiate between triangle and plane.
10. Solve numerical using cross product for normal.

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⚡ Quick Revision Summary

• Triangle = 3 connected points (always planar)

• Plane equation: [ Ax + By + Cz + D = 0 ]

• Normal vector: [ \vec{N} = (A, B, C) ]

• Found using: [ (P_2 - P_1) \times (P_3 - P_1) ]

• Used in:

  • 3D modeling
  • Shading
  • Rendering

• Key idea: Everything in 3D graphics is built from triangles

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