Project 10: The 3D Renderer (Linear Algebra & Projection)

Build a simple 3D renderer that projects points onto a 2D screen.

Project Overview

Attribute Value
Difficulty Level 2: Intermediate
Time Estimate 1-2 weeks
Main Language Python
Alternative Languages JavaScript, C++
Knowledge Area Linear algebra and projection
Tools Plotting tool
Main Book “Computer Graphics from Scratch” by Gabriel Gambetta

What you’ll build: A program that renders rotating 3D points or wireframes onto a 2D canvas using projection math.

Why it teaches math: Projection makes vectors and matrices practical. You can see transformations directly.

Core challenges you’ll face:

  • Implementing rotation matrices
  • Applying perspective projection
  • Managing camera position

Real World Outcome

You will render a rotating cube or simple model, showing a wireframe animation or sequence of frames.

Example Output:

$ python render.py --model cube
Rendered 120 frames
Saved frames to ./frames/

Verification steps:

  • Confirm rotations look correct
  • Compare projection results to expected perspective behavior

The Core Question You’re Answering

“How do 3D points become 2D pixels?”

This is the core math behind computer graphics.

Concepts You Must Understand First

Stop and research these before coding:

  1. Vectors and matrices
    • How does a matrix transform a vector?
    • Book Reference: “Computer Graphics from Scratch” by Gabriel Gambetta, Ch. 1
  2. Rotation matrices
    • How do you rotate a point around an axis?
    • Book Reference: “Linear Algebra and Its Applications” by Gilbert Strang, Ch. 5
  3. Perspective projection
    • Why do distant objects appear smaller?
    • Book Reference: “Computer Graphics from Scratch” by Gabriel Gambetta, Ch. 3

Questions to Guide Your Design

  1. Coordinate system
    • Where is the camera located?
    • Which axis is up?
  2. Rendering pipeline
    • Will you draw just points, or connect edges?
    • How will you handle depth ordering?

Thinking Exercise

Project a Point

Given a 3D point (2, 3, 4) and a camera at the origin, compute its projected 2D coordinates using a simple perspective rule.

Questions while working:

  • How does the z-value affect scale?
  • What happens if z approaches zero?

The Interview Questions They’ll Ask

Prepare to answer these:

  1. “What is a projection matrix?”
  2. “Why does rotation use matrices?”
  3. “What is the difference between orthographic and perspective projection?”
  4. “How do you handle depth ordering?”
  5. “What is a camera transform?”

Hints in Layers

Hint 1: Starting Point Render a single 3D point and project it to 2D.

Hint 2: Next Level Add rotation matrices to animate the point.

Hint 3: Technical Details Use homogeneous coordinates if you want to combine transforms.

Hint 4: Tools/Debugging Plot the projected points and check symmetry as you rotate.

Books That Will Help

Topic Book Chapter
Graphics pipeline “Computer Graphics from Scratch” by Gabriel Gambetta Ch. 1
Rotation matrices “Linear Algebra and Its Applications” by Gilbert Strang Ch. 5
Projection “Computer Graphics from Scratch” by Gabriel Gambetta Ch. 3

Implementation Hints

  • Start with a wireframe cube to keep it simple.
  • Keep projection parameters configurable.
  • Export frames to inspect motion step by step.

Learning Milestones

  1. First milestone: You can project a 3D point to 2D.
  2. Second milestone: You can rotate and render a cube.
  3. Final milestone: You can explain projection and camera transforms.