Chapter 1: Curve Theory

Analyzing the local properties of curves

Goal of this page

Define the curvature and torsion of a space curve and derive the Frenet–Serret formulas. Learn how to describe the local shape of a curve mathematically.

1. Regular curves and the arc-length parameter

Definition: regular curve

A curve $\gamma: I \to \mathbb{R}^3$ is regular when:

$$\gamma'(t) \neq 0 \quad (\forall t \in I)$$

Definition: arc-length parameter

Reparametrizing by the arc length $s(t) = \displaystyle\int_{t_0}^{t} |\gamma'(u)| \, du$ gives:

$$\left|\dfrac{d\gamma}{ds}\right| = 1$$

Such an $s$ is called the arc-length parameter.

2. Curvature

Definition: curvature

The curvature of a curve $\gamma(s)$ given in the arc-length parameter $s$:

$$\kappa(s) = \left|\dfrac{d^2\gamma}{ds^2}\right| = \left|\dfrac{d\mathbf{t}}{ds}\right|$$

Here $\mathbf{t} = d\gamma/ds$ is the unit tangent vector.

Intuitively, a circle of radius $R$ has curvature $\kappa = 1/R$, so the curvature measures how sharply the curve bends (the reciprocal of the radius of curvature $R$): a straight line has $\kappa = 0$, and tighter bends have larger $\kappa$.

P t R center of curvature
Figure 1: The osculating circle. At the point $P$ where the curvature is greatest, the circle is tangent to the curve and shares its curvature $\kappa$ (radius $R=1/\kappa$, with center at the center of curvature). The tangent $\mathbf{t}$ and the radius are perpendicular.

Theorem: formula for the curvature

For a curve $\gamma(t)$ given in a general parameter $t$:

$$\kappa = \dfrac{|\gamma' \times \gamma''|}{|\gamma'|^3}$$

For a plane curve $(x(t), y(t))$:

$$\kappa = \dfrac{|x'y'' - x''y'|}{(x'^2 + y'^2)^{3/2}}$$

3. The Frenet frame

Definition: the Frenet frame

An orthonormal basis carried along a space curve:

  • Tangent vector: $\mathbf{t} = \dfrac{d\gamma}{ds}$
  • Principal normal vector: $\mathbf{n} = \dfrac{1}{\kappa}\dfrac{d\mathbf{t}}{ds}$
  • Binormal vector: $\mathbf{b} = \mathbf{t} \times \mathbf{n}$
t n b P
Figure 2: The Frenet frame $\{\mathbf{t},\mathbf{n},\mathbf{b}\}$ at a point $P$ on a helix. $\mathbf{t}$ is the tangent, $\mathbf{n}$ the principal normal, and $\mathbf{b}=\mathbf{t}\times\mathbf{n}$ the binormal; $\mathbf{t}$ and $\mathbf{n}$ span the osculating plane.

4. Torsion and the Frenet–Serret formulas

Definition: torsion

The torsion measures how much the curve twists out of the osculating plane (the plane spanned by $\mathbf{t}$ and $\mathbf{n}$):

$$\tau = -\dfrac{d\mathbf{b}}{ds} \cdot \mathbf{n}$$

A curve lying in a plane (a line, circle, parabola, and so on) has $\tau = 0$; the torsion becomes nonzero only when the curve twists out of its osculating plane in three dimensions (for example, the helix below).

Theorem: the Frenet–Serret formulas

$$\dfrac{d}{ds}\begin{pmatrix} \mathbf{t} \\ \mathbf{n} \\ \mathbf{b} \end{pmatrix} = \begin{pmatrix} 0 & \kappa & 0 \\ -\kappa & 0 & \tau \\ 0 & -\tau & 0 \end{pmatrix} \begin{pmatrix} \mathbf{t} \\ \mathbf{n} \\ \mathbf{b} \end{pmatrix}$$

Theorem: the fundamental theorem of curves

Given a curvature $\kappa(s) > 0$ and a torsion $\tau(s)$, there exists a space curve, unique up to a rigid motion, having them as its curvature and torsion.

Example: the helix

The curvature and torsion of $\gamma(t) = (a\cos t, a\sin t, bt)$:

$$\kappa = \dfrac{a}{a^2 + b^2}, \quad \tau = \dfrac{b}{a^2 + b^2}$$

Both are constant, so a helix bends and twists uniformly.

Summary

Key points of this chapter

  • Curvature $\kappa$: how sharply the curve bends (the reciprocal of the radius of curvature)
  • Torsion $\tau$: how much the curve twists
  • Frenet frame: the orthonormal basis $\{\mathbf{t},\mathbf{n},\mathbf{b}\}$
  • Fundamental theorem: $\kappa$ and $\tau$ determine the curve up to a rigid motion

FAQ

Q1. What do the Frenet–Serret formulas describe?

The Frenet–Serret formulas describe how the tangent $\mathbf{t}$, principal normal $\mathbf{n}$, and binormal $\mathbf{b}$ of a regular curve change with respect to arc length $s$: $\mathbf{t}'=\kappa\mathbf{n}$, $\mathbf{n}'=-\kappa\mathbf{t}+\tau\mathbf{b}$, $\mathbf{b}'=-\tau\mathbf{n}$. The curvature $\kappa$ (with $\kappa > 0$) and torsion $\tau$ determine the curve completely up to a rigid motion, under suitable regularity assumptions.

Q2. What is the advantage of using the arc-length parameter?

With the arc-length parameter $s$ the speed satisfies $|\gamma'(s)|=1$, so the curvature is obtained directly as $\kappa=|\gamma''(s)|$. For a general parameter $t$ one instead needs the more cumbersome formula $\kappa=|\gamma'\times\gamma''|/|\gamma'|^3$. The arc-length parameter is the standard choice for discussing the intrinsic properties of a curve.

Q3. What is the turning number of a closed curve?

The turning number (rotation index) of a plane closed curve is the total angle through which the tangent vector rotates during one loop, divided by $2\pi$. For a simple closed curve (no self-intersections) it is $\pm 1$ (Hopf's Umlaufsatz), with the sign depending on orientation. It is a topological invariant and a starting point for knot theory.

References