📐 Mathematics Foundations for Computer Graphics

Computer graphics is often described as a visual field, but at its core, it is deeply mathematical. Every image we render is the result of geometric reasoning, algebraic transformations, and numerical computation.

Rather than treating mathematics as a collection of formulas, it is more useful to view it as a language for describing shape, motion, light, and space. This page summarizes the key mathematical tools that appear repeatedly across graphics, rendering, and simulation.

💪 Linear Algebra

Linear algebra is the foundation of almost all geometric computation in computer graphics. Vectors represent positions, directions, and colors, while matrices represent transformations between coordinate systems.

Common operations such as translation, rotation, scaling, and projection can all be expressed as matrix multiplications. This unified representation allows complex transformations to be composed efficiently.

• Vectors: position, direction, normals
• Matrices: transformations and coordinate changes
• Dot and cross products: angles, projections, orientation

⚪ Geometry

Geometry describes the shape and structure of objects in space. In graphics, we work with both discrete geometry (meshes and polygons) and continuous geometry (curves, surfaces, and implicit shapes).

Many practical problems — collision detection, ray–surface intersection, and visibility — reduce to geometric queries such as distances, angles, and intersections.

• Points, lines, planes, and triangles
• Curves and surfaces (parametric and implicit)
• Distance, intersection, and orientation tests

🍰 Calculus

Calculus enters graphics whenever we reason about change, motion, or accumulation. Derivatives describe how quantities vary, while integrals describe how light, mass, or probability accumulates over space and time.

In rendering, calculus appears naturally in the rendering equation. In animation and simulation, it governs motion, forces, and energy.

• Derivatives: gradients, normals, and optimization
• Integrals: light transport and accumulation
• Differential equations: motion and dynamics

🎲 Probability & Statistics

Modern rendering relies heavily on probability. When exact computation is infeasible, random sampling provides an efficient approximation. This idea underlies Monte Carlo rendering.

Instead of computing light exactly, we estimate it statistically by sampling directions, surfaces, or paths. Understanding variance and convergence is therefore crucial.

• Random variables and distributions
• Expectation and variance
• Monte Carlo integration

🧮 Numerical Methods

Computer graphics is ultimately implemented on finite-precision machines. Numerical methods help us design algorithms that are stable, efficient, and robust to error.

Many failures in graphics — flickering, exploding simulations, or noisy renders — are numerical issues rather than conceptual ones.

• Floating-point precision
• Iterative solvers and approximation
• Stability and convergence

🌼 Summary

Mathematics provides the structure behind visual realism. Linear algebra shapes space, geometry defines form, calculus models change, probability handles uncertainty, and numerical methods ensure robustness.

A strong intuition for these tools makes it easier to move between theory, implementation, and visual results — which is the essence of computer graphics.