Math for Game Programmers

Points, displacement, and coordinate conventions

A point describes a position in a coordinate system; a vector describes displacement or direction. Subtracting points produces a vector, while adding a displacement to a point produces another point. The vector (3, 4, 0) has length 5, so its unit direction is (0.6, 0.8, 0). Normalize only when length is nonzero. Keep units consistent: multiplying a velocity measured in metres per second by seconds produces metres. State which axis is up and whether the coordinate system is left-handed or right-handed before interpreting rotations.

References: COMP 1008 — Fall 2026 course outline · Godot Engine — Math tutorials

Dot products, cross products, and normals

The dot product measures alignment: a dot b = |a||b| cos(theta). With unit vectors it equals the cosine of the angle, making it useful for field-of-view and facing tests. The cross product produces a perpendicular vector with magnitude equal to the parallelogram area. Its sign depends on operand order and the coordinate convention. A surface normal describes orientation rather than position. For nonuniform scaling, transforming normals like ordinary directions can make them cease to be perpendicular; use the inverse transpose of the linear transform and renormalize.

References: COMP 1008 — Fall 2026 course outline · Godot Engine — Math tutorials

Matrices, homogeneous coordinates, and inverse transforms

A transform combines scale, rotation, and translation. Homogeneous coordinates allow translation to join matrix multiplication: points use w = 1, directions use w = 0. With column vectors, the rightmost matrix acts first. Therefore translate-then-rotate generally differs from rotate-then-translate. An inverse maps world coordinates back to a local frame when the transform is invertible. A camera view matrix is conceptually the inverse of the camera’s world transform. A zero scale destroys information, so the corresponding transform cannot have a full inverse.

References: COMP 1008 — Fall 2026 course outline · Godot Engine — Math tutorials

Planes, bounds, and conservative tests

A plane can be expressed as n dot x + d = 0. For a unit normal, this expression gives signed perpendicular distance. A sphere with centre c and radius r lies wholly on the positive side when n dot c + d > r. Bounding boxes and spheres provide cheap broad-phase rejection before costly exact geometry checks. A view frustum is a collection of clipping half-spaces. Conservative culling may keep an invisible object, but it should not discard visible geometry; assess the effect of numerical tolerances near boundaries.

References: COMP 1008 — Fall 2026 course outline · Godot Engine — Math tutorials

Sweeping and a worked collision calculation

Discrete collision tests examine isolated moments and can miss a fast object crossing a thin obstacle. A sweep examines the path over an interval. Suppose a point starts at x = 1, moves at 6 units per second, and approaches the plane x = 4. Solve 1 + 6t = 4 to obtain t = 0.5 seconds. A radius-0.5 sphere touches the same plane when its centre reaches 3.5, at t = 2.5/6 seconds. Check the valid time interval and whether the object is moving toward the surface.

References: COMP 1008 — Fall 2026 course outline · Godot Engine — Math tutorials

Interpolation, billboards, and quaternions

Linear interpolation combines endpoints as (1 - t)a + tb for t between zero and one. It is useful for positions but does not generally preserve rotation constraints. Unit quaternions represent rotations compactly and can interpolate orientation smoothly. The quaternions q and -q represent the same orientation, so interpolation often chooses signs to use the shorter arc. Quaternion multiplication composes rotations and is order-sensitive. A billboard constructs an orientation that faces a camera; choose a stable fallback when the desired facing direction becomes parallel to the chosen up vector.

References: COMP 1008 — Fall 2026 course outline · Godot Engine — Math tutorials

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.

• COMP 1008 — Fall 2026 course outline ↗Course scope · Fall 2026
• Godot Engine — Math tutorials ↗Primary reference · Public reference; checked 2026-11-05