Foundations of Game Programming and Computer Graphics

Vectors, coordinate frames, and orientation

A point identifies a location; a vector represents a displacement or direction. The dot product relates directions and supports projection, while the cross product produces an oriented perpendicular vector in three dimensions. Normalize when a formula assumes unit length, and handle the zero vector explicitly. Coordinates only have meaning relative to a frame. A direction expressed in object coordinates cannot be compared directly with a light direction expressed in world coordinates without a transformation.

References: Physically Based Rendering — geometry, transforms, cameras, textures

Transformations and hierarchy

Homogeneous coordinates express translation, rotation, and scale using matrices. Matrix multiplication is generally noncommutative, so the order of operations is part of the model. A child object's world transformation combines its local transformation with its parent's world transformation under the selected vector convention. Camera view transformations convert world coordinates into the camera frame; projection then maps visible geometry toward screen coordinates. Keep conventions explicit, including handedness, axis directions, multiplication order, and the meaning of depth.

References: Physically Based Rendering — geometry, transforms, cameras, textures

Worked hierarchical transformation

Using column vectors, let a child's local point be (1,0,0,1). Scale by two and then translate by three on x: T*S*p gives x=5. Reversing the order gives S*T*p with x=8 because the translation is also scaled. A robot arm exhibits the same issue: translating to a joint and rotating around it differs from rotating the whole arm around the world origin. Compute a simple point by hand to debug a complicated matrix chain.

References: Physically Based Rendering — geometry, transforms, cameras, textures

Meshes, normals, and texture coordinates

A triangle mesh represents a surface through vertices and connectivity. Vertex positions, normals, colors, and texture coordinates may require different values across seams even at a shared geometric location. A surface normal transforms with the inverse transpose of the linear transformation when nonuniform scaling is present. Texture coordinates map a surface to image samples; filtering and mipmaps reduce aliasing when many texels map to one pixel. A normal map changes local shading directions without adding the corresponding geometric silhouette.

References: Physically Based Rendering — geometry, transforms, cameras, textures

The graphics pipeline and illumination

A typical raster pipeline transforms vertices, assembles primitives, clips them, determines covered fragments, shades them, and applies depth and blending operations. Interpolation supplies fragment attributes, with perspective correction needed for quantities such as texture coordinates. Diffuse illumination depends on the angle between the surface normal and light direction; a specular term depends on viewing geometry and material assumptions. A shading model approximates appearance and should be evaluated under its assumptions rather than mistaken for a universal physical law.

References: Physically Based Rendering — geometry, transforms, cameras, textures

Quaternions, particles, and practical debugging

Unit quaternions represent orientations compactly and avoid some difficulties of Euler-angle parameterizations. Composition order still matters, and interpolation should respect orientation geometry. A particle system updates many simple entities with state such as position, velocity, lifetime, and color. Separate simulation time from rendering rate when frame variation would otherwise change motion. Debug graphics incrementally: verify coordinates, render simple colors, visualize normals, disable effects, and test one transform at a time before combining lighting, textures, animation, and shadows.

References: Physically Based Rendering — geometry, transforms, cameras, textures

Practice & worked answers

Original exercises for this guide.

Public notes & references

Course sources establish the scope above. Reference materials support the explanations; linked lecture notes and books remain with their authors.

• Carleton COMP 3501 — Fall 2026 outline ↗Course scope · Fall 2026
• Physically Based Rendering — geometry, transforms, cameras, textures ↗Open author textbook
• Glenn Fiedler — Fix Your Timestep ↗Author technical article