Computer graphics
Normals, lighting, curves, ray tracing and rendering: derivatives orient surfaces, interpolation fills triangles and integrals gather light.
8 topics
Rendering an image asks two calculus questions per pixel. Geometry: where does the ray hit, and which way does the surface face there? The answer is a derivative — the normal is the gradient of an implicit surface or the cross product of the partial derivatives of a parametric one. Light: how much arrives at that point from every direction? The answer is an integral over the hemisphere, the rendering equation, estimated by Monte Carlo. Curves and surfaces for design are polynomials glued with continuity conditions on their derivatives.
Topics
Surface normals
The unit vector perpendicular to a surface at a point. For an implicit surface it is ; for a parametric one , the normalized cross product . Every lighting computation starts from it.
Lighting and shading
Computing the colour of a surface point from the light, the normal and the viewer. Lambertian diffuse light is ; specular highlights use . Values are interpolated across triangles.
Bézier curves and splines
Polynomial curves controlled by a few points, glued with continuity conditions on derivatives (: same tangent, : same curvature). Fonts, SVG paths, CAD surfaces and animation curves are built from them.
Procedural generation and noise
Generating terrain, clouds and textures from smooth random functions (Perlin noise) summed over octaves (fractal Brownian motion). Smooth interpolants (, with zero first and second derivatives at the ends) avoid visible grid artefacts; analytic derivatives give normals for lighting.
Ray tracing
Follow rays of light from the camera into the scene: solve for intersections (roots of equations), reflect and refract using the normal, and recurse. Real-time ray tracing is now in consumer GPUs.
The rendering equation
Kajiya (1986): the light leaving a point equals the light it emits plus the integral, over all incoming directions, of incoming light times the material's reflectance times a cosine. An integral equation solved by Monte Carlo path tracing — the method behind modern film rendering.
Signed distance fields and ray marching
Represent a shape by the signed distance to its surface. Rays march forward by (it is always safe), the surface is , and the normal is . Neural SDFs learn with a network.
Mesh processing (discrete differential geometry)
Calculus on triangle meshes: a discrete Laplacian (cotangent weights), discrete curvatures and integrals over faces. Used for smoothing scanned models, parametrizing for textures, deforming characters and computing geodesics.
The mathematics this domain runs on
ƒ Foundations ★★★★★
∇ Multivariable calculus ★★★★★
- Surfaces and level sets★★★★★→Surface normals, Signed distance fields and ray marching
- Gradient★★★★★→Surface normals, Signed distance fields and ray marching
- Curvature (basic differential geometry)★★★★★→Bézier curves and splines, Mesh processing (discrete differential geometry)
- Multiple integrals and change of variables★★★★★→The rendering equation