ISEGORIA / MATH ENCYCLOPEDIA
Differential geometry: curvature and shape
Metrics, geodesics, and curvature on surfaces.
Before you begin: Calculus and vectors
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. Curvature of a curve
Choose a plane curve and how strongly it bends, then drag a point along it. The osculating circle is the circle that fits the curve best at that point: its radius is 1/|κ|. The curvature comb draws κ as spikes along the curve, and the graph on the right plots signed curvature against arc length, positive where the curve turns left and negative where it turns right.
Worked example. A circle of radius \(R\) has constant curvature \(1/R\).
Watch out. The formula gives signed curvature; its absolute value is the curvature of the curve as an unoriented set, and the sign flips if the curve is traversed the other way.
Where is \(\kappa\) largest?
Where the tangent direction changes fastest with arc length.
2. Geodesics on a torus
Launch a geodesic on a torus from a point on the tube at angle v₀, heading at angle ψ to the parallel, and compare it with the coordinate line that starts the same way. The geodesic is computed from the geodesic equation with the torus's Christoffel symbols. In the flat (u, v) chart the coordinate line is straight but the geodesic bends; on the surface it is the other way round. Clairaut's relation ρ cos ψ = const decides whether the geodesic winds through the hole or stays on the outside.
Worked example. With R = 2 and r = 0.8, a geodesic starting on the outer equator with ψ = 30° has c = 2.8 cos 30° ≈ 2.42 > R − r = 1.2, so it oscillates about the outer equator and never reaches the inner one.
Watch out. The torus picture uses flat shading and marks the part of each curve on the far side, as seen from the viewer, with dashes; it is a projection, not a rendering of true occlusion.
What does the connection correct?
It subtracts coordinate artefacts from ordinary acceleration.
3. Gaussian curvature
The graph z = ½(x² + b y²) morphs from a bowl (b > 0) through a parabolic cylinder (b = 0) to a saddle (b < 0). Drag the gold point over the curvature map on the right: the readout computes the first and second fundamental forms there and K = (LN − M²)/(EG − F²). Drag the surface on the left to turn it.
Worked example. At the origin E = G = 1, F = M = 0, L = 1 and N = b, so K = b: a sphere-like bowl for b > 0 and a saddle with K < 0 for b < 0. Away from the origin K = b/(1 + x² + b²y²)² shrinks toward 0.
Watch out. K is intrinsic, but L, M and N are not: they depend on how the surface sits in space. Only the combination LN − M² divided by EG − F² is unchanged by bending without stretching.
Can bending without stretching change \(K\)?
For an isometric bend, intrinsic Gaussian curvature is preserved.