Curves in the plane and in space

Differential geometry studies shape through derivatives. For curves the whole local story is two functions, curvature and torsion; the global story ties curvature to topology.

1. Regular curves and arc length (Def. 1.1); curvature for any parametrisation (Prop. 1.2).
2. The Frenet–Serret equations (Thm 1.3), torsion (Prop. 1.4), planarity (Cor. 1.5), and the fundamental theorem of curves (Thm 1.6).
3. Signed curvature of plane curves (Def. 1.7) and the rotation index (Thm 1.8).
4. Global results: Wirtinger's inequality (Lemma 1.9), the isoperimetric inequality (Thm 1.10) and the four vertex theorem (Thm 1.11).

A smooth curve $\gamma: I\to\mathbb{R}^3$ is regular if $\gamma'(t)\ne0$ for all $t$. Its arc length from $t_0$ is $s(t) = \int_{t_0}^t|\gamma'(u)|\,\mathrm{d}u$; since $s' = |\gamma'| > 0$, $s$ is invertible and $\gamma$ can be reparametrised to have unit speed, $|\gamma'(s)| = 1$. Length is unchanged by reparametrisation.

For a unit-speed curve the curvature is $\kappa(s) = |\gamma''(s)|$. For a regular curve in any parametrisation,

Let $\gamma$ be unit speed with $\kappa > 0$. Set $\mathbf{T} = \gamma'$, $\mathbf{N} = \mathbf{T}'/\kappa$, $\mathbf{B} = \mathbf{T}\times\mathbf{N}$, an orthonormal frame. There is a function $\tau$, the torsion, with

For a regular curve with $\gamma'\times\gamma''\ne0$,

For the helix $(a\cos t, a\sin t, bt)$ with $a > 0$: $\kappa = \frac{a}{a^2 + b^2}$ and $\tau = \frac{b}{a^2 + b^2}$.

A unit-speed curve with $\kappa > 0$ lies in a plane if and only if $\tau\equiv0$. A curve with $\kappa\equiv0$ is (part of) a straight line.

Given smooth $\kappa > 0$ and $\tau$ on an interval $I$, there is a unit-speed curve with curvature $\kappa$ and torsion $\tau$, and any two such curves differ by a rigid motion (a rotation followed by a translation).

For a unit-speed plane curve write $\mathbf{T} = (\cos\theta, \sin\theta)$ with $\theta(s)$ a smooth turning angle, and let $\mathbf{N}_s$ be $\mathbf{T}$ rotated anticlockwise by $\frac\pi2$. The signed curvature is $\kappa_s = \theta'$, so $\mathbf{T}' = \kappa_s\mathbf{N}_s$ and $|\kappa_s| = \kappa$. In a general parametrisation $\kappa_s = \frac{x'y'' - x''y'}{(x'^2 + y'^2)^{3/2}}$.

For a closed unit-speed plane curve of length $L$, $\int_0^L\kappa_s\,\mathrm{d}s = \theta(L) - \theta(0) = 2\pi n$ for an integer $n$, the rotation index. If the curve is simple (no self-intersections), $n = \pm1$: total signed curvature is $\pm2\pi$.

If $f$ is $C^1$ and $2\pi$-periodic with $\int_0^{2\pi}f = 0$, then $\int_0^{2\pi}f'^2\ge\int_0^{2\pi}f^2$, with equality iff $f = a\cos t + b\sin t$.

A simple closed plane curve of length $L$ enclosing area $A$ satisfies $L^2\ge4\pi A$, with equality iff it is a circle.

A vertex of a plane curve is a critical point of $\kappa_s$. Every simple closed convex curve that is not a circle has at least four vertices. (The theorem holds for all simple closed curves; we prove the convex case.)

• In unit speed use $\kappa = |\gamma''|$; otherwise $\kappa = |\gamma'\times\gamma''|/|\gamma'|^3$ (never $|\gamma''|$).
• $\tau = \det(\gamma', \gamma'', \gamma''')/|\gamma'\times\gamma''|^2$; quick test: $\tau = 0$ everywhere ⇔ planar.
• Plane curves: use signed $\kappa_s$; a closed simple curve has $\int\kappa_s = \pm2\pi$.
• Trap: Frenet frames need $\kappa > 0$.