📘 Triangles and Planes — Exam Notes (Computer Graphics)
🔹 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.
🔹 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)
]
✔️ Why Triangles are Important?
- Always planar (lie on a single plane)
- Simple to compute
- Stable for rendering
- Used to build complex 3D models
✔️ Triangle Representation
A triangle has:
- 3 vertices
- 3 edges
- 1 surface
🔹 3. Plane in Computer Graphics
✔️ Definition
A plane is a flat 2D surface extending infinitely in 3D space.
✔️ General Equation of a Plane
[
Ax + By + Cz + D = 0
]
Where:
- (A, B, C) = normal vector components
- (D) = constant
🔹 4. Normal Vector of a Plane (Very Important)
✔️ Definition
A normal vector is a vector perpendicular to the plane.
[
\vec{N} = (A, B, C)
]
✔️ Finding Normal from Triangle
Given points (P_1, P_2, P_3):
-
Find two vectors:
[
\vec{U} = P_2 - P_1,\quad \vec{V} = P_3 - P_1
]
-
Compute cross product:
[
\vec{N} = \vec{U} \times \vec{V}
]
👉 This gives the plane’s normal vector.
🔹 5. Equation of Plane Using Three Points
✔️ Steps
- Take three points of triangle
- Find two direction vectors
- Compute cross product → normal vector
- Substitute into plane equation
🔹 6. Triangle Normal (Shading Use)
✔️ Importance
- Used in lighting and shading
- Determines how light reflects from surface
✔️ Formula
[
\vec{N} = (P_2 - P_1) \times (P_3 - P_1)
]
🔹 7. Plane Classification
✔️ 7.1 Front Face / Back Face
- Determined using normal vector direction
If:
- Normal points toward viewer → Front face
- Away from viewer → Back face
✔️ 7.2 Back-Face Culling (Important)
- Removes faces not visible to camera
- Improves rendering speed
🔹 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
🔹 9. Distance of Point from Plane
✔️ Formula
[
d = \frac{|Ax + By + Cz + D|}{\sqrt{A^2 + B^2 + C^2}}
]
🔹 10. Plane in Rendering
Planes are used in:
- Polygon surfaces
- 3D object modeling
- Collision detection
- Lighting calculations
🔹 11. Triangles in 3D Modeling
✔️ Triangle Mesh
🔹 12. Advantages of Triangles
- Always planar
- Easy to compute
- Efficient rendering
- Works well with GPUs
🔹 13. Limitations
- Large models require many triangles
- Can increase memory usage
🔹 14. Diagram Descriptions
✔️ Triangle in 3D
- Draw 3 points in space
- Connect them to form triangle
✔️ Plane
- Draw large flat surface
- Show normal vector perpendicular to surface
✔️ Cross Product
- Show two vectors forming a plane
- Resulting normal vector perpendicular
📝 Likely Exam Questions
- Define triangle in computer graphics.
- Why are triangles important in 3D modeling?
- Write equation of a plane.
- How do you find normal vector of a plane?
- Explain plane equation using three points.
- What is back-face culling?
- Derive distance from point to plane.
- Explain triangle mesh.
- Differentiate between triangle and plane.
- Solve numerical using cross product for normal.
⚡ 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