Week 17 — Differential Geometry of Curves and Surfaces

Overview

Differential geometry is the mathematics of smooth shape: how curves bend and twist, how surfaces curve, and what “straight line” means on a curved surface. It underlies trajectory geometry, road and camera modeling, and the geometry of configuration spaces in robotics. This week works through curves (arc length, curvature, torsion, the Frenet frame), surfaces and tangent planes, the first and second fundamental forms, the curvatures they define, Gauss’s Theorema Egregium, and geodesics — the shortest/straightest paths.

Readings

Pressley Ch 1 — Curves. Parametrized curves, arc length, curvature, torsion, the Frenet–Serret formulas.
Pressley Ch 4 — Surfaces in three dimensions. Smooth surfaces, parametrizations, tangent planes, normal vectors.
Pressley Ch 5 — The first fundamental form. Lengths, angles, and areas on a surface.
Pressley Ch 6 — Curvature of surfaces. The second fundamental form, normal/principal/Gaussian/mean curvature.
Pressley Ch 8 — Geodesics. The geodesic equations and shortest paths.

Key Concepts

Curves: arc length, curvature, torsion

A curve \(\boldsymbol{\gamma}(t)\) is reparametrized by arc length \(s\) so that \(\|\boldsymbol{\gamma}'(s)\| = 1\) (unit speed). The Frenet frame \((\mathbf{T}, \mathbf{N}, \mathbf{B})\) — tangent, normal, binormal — evolves by the Frenet–Serret formulas:

\[\mathbf{T}' = \kappa \mathbf{N}, \quad \mathbf{N}' = -\kappa\mathbf{T} + \tau\mathbf{B}, \quad \mathbf{B}' = -\tau\mathbf{N}.\]

The curvature \(\kappa\) measures bending in the osculating plane; the torsion \(\tau\) measures how the curve twists out of it. Curvature is exactly the quantity that limits feasible vehicle paths (steering radius) and defines smooth trajectories.

Surfaces and tangent planes

A regular surface is locally the image of a smooth parametrization \(\boldsymbol{\sigma}(u,v)\) whose partials \(\boldsymbol{\sigma}_u, \boldsymbol{\sigma}_v\) are independent; they span the tangent plane, and their normalized cross product is the unit normal \(\mathbf{n}\). This is the setting for any 3-D surface model — terrain, car bodies, lens surfaces.

First fundamental form

The first fundamental form is the restriction of the ambient dot product to the tangent plane:

\[\mathrm{I} = E\,du^2 + 2F\,du\,dv + G\,dv^2, \qquad E = \boldsymbol{\sigma}_u\!\cdot\!\boldsymbol{\sigma}_u,\ F = \boldsymbol{\sigma}_u\!\cdot\!\boldsymbol{\sigma}_v,\ G = \boldsymbol{\sigma}_v\!\cdot\!\boldsymbol{\sigma}_v.\]

It is the intrinsic metric: lengths of curves, angles, and areas on the surface are computed from \(E,F,G\) alone, without reference to how the surface sits in space.

Second fundamental form and curvatures

The second fundamental form measures how the surface bends away from its tangent plane (how the normal turns). Its eigenvalues relative to the first fundamental form are the principal curvatures \(\kappa_1, \kappa_2\), from which:

\[K = \kappa_1\kappa_2 \ \text{(Gaussian curvature)}, \qquad H = \tfrac12(\kappa_1+\kappa_2) \ \text{(mean curvature)}.\]

This is again a (generalized) eigenvalue problem (Week 2). \(K>0\) is dome-like, \(K<0\) saddle-like, \(K=0\) flat/developable.

Theorema Egregium and geodesics

Gauss’s Theorema Egregium: the Gaussian curvature \(K\) is intrinsic — computable from the first fundamental form alone — even though it is defined via the extrinsic second form. Consequence: a flat map cannot preserve distances on a sphere (\(K\ne 0\)), the root cause of all map-projection distortion. A geodesic is a locally shortest / “straightest” path, with zero tangential acceleration; it satisfies the second-order geodesic equations and generalizes the straight line to curved surfaces — the shortest-path notion behind motion on manifolds.

Connections

Backward: geodesics and configuration-space geometry generalize the Lagrangian dynamics of Week 16.
Across courses: trajectory curvature and road geometry (Courses 1, 2), camera/lens surface geometry and image formation (Course 2 sensors, Week 18 optics).