Part IX — The Mathematics of Real-Time Rendering

Booklet: Abridged Math Foundations for Signal Processing, Machine Learning, and Artificial Intelligence, Lessons 63–69 (PDF).

Points, vectors, and homogeneous coordinates — affine maps, the view matrix, perspective projection and its hyperbolic depth, reverse-Z; rotations — \(SO(3)\), Rodrigues’ formula, Euler angles and gimbal lock, quaternions and slerp; triangles, planes, and frusta — barycentric coordinates, edge functions, perspective-correct interpolation, frustum-plane extraction and culling, the geometry of shadow maps; radiometry and the rendering equation — the Lambertian \(1/\pi\), Blinn–Phong, Fresnel, and the Cook–Torrance microfacet model with GGX; integrating light — Monte Carlo shading, cosine and GGX importance sampling, the split sum, spherical harmonics, mipmapping as prefiltering; color — spectra as vectors, metamers, sRGB, high dynamic range, and tone mapping; Gaussian splatting — anisotropic Gaussians, their linearized projection, and alpha compositing.

28 exercises across Lessons 63–69 — 14 [Hand], 14 [Proof]. The theory that Course 4 — Real-Time Rendering & GPU Engineering applies and does not re-derive.

Exercises

Worked sets are linked below as they are completed.

  • Lesson 63 — Points, Vectors, and Homogeneous Coordinates: Affine Maps, Cameras, and Perspective Projection — 4 exercises (2 [Hand], 2 [Proof])
  • Lesson 64 — Rotations and Orientation: \(SO(3)\), Axis–Angle, Euler Angles, and Quaternions — 4 exercises (2 [Hand], 2 [Proof])
  • Lesson 65 — Triangles, Planes, and Frusta: Barycentric Coordinates, Rasterization, Interpolation, and Culling — 4 exercises (2 [Hand], 2 [Proof])
  • Lesson 66 — Radiometry, the Rendering Equation, and Reflection Models: Lambert, Blinn–Phong, Fresnel, and Microfacets — 4 exercises (2 [Hand], 2 [Proof])
  • Lesson 67 — Integrating Light: Monte Carlo Shading, Importance Sampling, Spherical Harmonics, and the Split Sum — 4 exercises (2 [Hand], 2 [Proof])
  • Lesson 68 — Color and the Display: Spectra as Vectors, Linear Color, sRGB, High Dynamic Range, and Tone Mapping — 4 exercises (2 [Hand], 2 [Proof])
  • Lesson 69 — Gaussian Splatting: Anisotropic Gaussians, Their Projection, and Alpha Compositing — 4 exercises (2 [Hand], 2 [Proof])