Curvature and Torsion of Space Curves

Chapter 7: Differential Geometry — Curvature

Learning objectives

Compute curvature $\kappa$ and torsion $\tau$ for a parametrized space curve
Construct the Frenet-Serret frame $(\mathbf{T},\mathbf{N},\mathbf{B})$ at a point and interpret each vector geometrically
State and apply the Frenet-Serret formulas
Recognise the fundamental theorem of space curves: $\kappa(s)$ and $\tau(s)$ determine the curve up to rigid motion

In the plane a curve can only bend; in space it can also twist. A helix climbing up a staircase, a strand of DNA, a roller-coaster track — all of these have shape that no single scalar $\kappa$ can encode. Differential geometry handles this with a second invariant, the torsion $\tau$ , measuring how fast the curve twists out of its instantaneous plane of bending. Together $\kappa$ and $\tau$ determine the shape of any smooth space curve uniquely (up to rigid motion). That uniqueness is the fundamental theorem of space curves.

The Frenet-Serret frame

For a smooth curve $\mathbf{r}(s)$ parametrized by arc length and with $\kappa(s) > 0$ , define three mutually orthogonal unit vectors at each point:

Unit tangent $\mathbf{T}(s) = \mathbf{r}'(s)$ — points in the direction of motion. Length $1$ because $s$ is arc length.
Principal normal

\mathbf{N}(s) = \mathbf{T}'(s)/|\mathbf{T}'(s)|

— the direction in which $\mathbf{T}$ is currently rotating, perpendicular to $\mathbf{T}$ . Points toward the centre of the osculating circle.

Binormal

\mathbf{B}(s) = \mathbf{T}(s) \times \mathbf{N}(s)

— perpendicular to both, completing a right-handed orthonormal triple.

The plane spanned by $\mathbf{T}$ and $\mathbf{N}$ is the osculating plane — the instantaneous plane of bending. If this plane never changes (constant $\mathbf{B}$ ), the curve is planar.

The Frenet-Serret formulas

The three frame vectors evolve along the curve according to a tightly coupled system:

\mathbf{T}'(s) = \kappa(s)\, \mathbf{N}(s)

\mathbf{N}'(s) = -\kappa(s)\, \mathbf{T}(s) + \tau(s)\, \mathbf{B}(s)

\mathbf{B}'(s) = -\tau(s)\, \mathbf{N}(s)

The first formula defines $\kappa$ (the rate at which the tangent rotates inside the osculating plane). The third defines $\tau$ (the rate at which the binormal — and therefore the osculating plane itself — rotates about $\mathbf{T}$ ). The middle formula is forced by orthonormality of the frame.

The arbitrary-parameter formulas

For a curve $\mathbf{r}(t)$ in any parametrization, the invariants are:

\kappa(t) = \dfrac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3}, \qquad \tau(t) = \dfrac{(\mathbf{r}'(t) \times \mathbf{r}''(t)) \cdot \mathbf{r}'''(t)}{|\mathbf{r}'(t) \times \mathbf{r}''(t)|^2}

For the standard helix $\mathbf{r}(t) = (a\cos t, a\sin t, bt)$ with $a > 0$ , both invariants are constant: $\kappa = a/(a^2 + b^2)$ and $\tau = b/(a^2 + b^2)$ . A circle is the special case $b = 0$ : $\kappa = 1/a$ and $\tau = 0$ .

The fundamental theorem

Given any smooth positive function $\kappa(s)$ and any smooth function $\tau(s)$ defined on an interval, there exists a smooth space curve realising them, unique up to a rigid motion of $\mathbb{R}^3$ . In other words, the pair $(\kappa, \tau)$ is a complete set of local invariants for space curves: knowing them is knowing the curve, modulo translation and rotation.

Where this shows up

- **DNA and biopolymer modelling:** The DNA double helix has a baseline torsion of roughly

10^{-1}

per base pair, and supercoiled DNA stores energy by deviating from that torsion. Models in molecular biology use Frenet frames along the backbone to compute writhe, link number, and the energetics of transcription bubbles. - **Roller-coaster engineering:** A roller-coaster track is a space curve whose

\kappa

and

\tau

profiles determine the g-forces felt by riders. Designers tune the curvature to keep accelerations within human tolerance and the torsion to avoid sudden head-snapping rolls. Real designs are evaluated by plotting both invariants along the arc length. - **Computer-aided surgery:** Catheter and endoscope navigation systems track the instrument tip as a space curve. Real-time computation of

\kappa

and

\tau

from a sequence of position samples lets the software warn the surgeon when the device is approaching a kink that risks vessel damage. - **Particle physics tracking:** A charged particle in a magnetic field follows a helix; reconstructing the helix radius (which gives

\kappa

) and pitch (which gives

\tau

) from detector hits is how the ATLAS and CMS experiments measure momentum and charge sign at the LHC. - **Computer graphics, sweep surfaces:** Tools that sweep a profile along a 3D path (CAD pipes, electrical conduit, animation hair) use the Frenet frame to orient the profile. A pure Frenet frame "flips" at inflection points, so production code typically uses the rotation-minimising frame — itself a corrected Frenet construction.

Pause and think: Why does the principal normal $\mathbf{N}$ become undefined at an inflection point of a space curve? (Hint: at an inflection $\mathbf{T}'(s) = \mathbf{0}$ .) What does that singularity tell you about the limits of the Frenet machinery, and how would you fix it in code?

Try it

Predict first: for the helix $\mathbf{r}(t) = (\cos t, \sin t, t)$ , is the torsion positive or negative? Compute it with the cross-product formula and verify your intuition about right-handedness.
For the planar curve $\mathbf{r}(t) = (t, t^2, 0)$ , compute $\tau$ . The answer should be $0$ — explain in one sentence why every planar curve embedded in $\mathbb{R}^3$ has zero torsion.
True or false: a curve with $\kappa = \tau = 0$ everywhere is a straight line. Justify directly from the Frenet-Serret formulas.
The unit-speed Frenet-Serret formulas can be written as a matrix ODE:

\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}

. Verify that the coefficient matrix is skew-symmetric, and explain why that guarantees the frame stays orthonormal.

A trap to watch for

The Frenet-Serret formulas are written for arc-length parametrization. When you use them with a non-unit-speed $\mathbf{r}(t)$ , you must remember that derivatives w.r.t. $t$ are not the same as derivatives w.r.t. $s$ : $d/ds = (1/|\mathbf{r}'(t)|) , d/dt$ . The cross-product formulas above already include this scaling, but students often try to plug $d\mathbf{r}/dt$ directly into $\mathbf{T}'(s) = \kappa\mathbf{N}$ and produce wrong answers. Rule: either re-parametrize by arc length first, or use the explicit- $t$ cross-product formulas.

What you now know

You can extract two intrinsic scalars — curvature $\kappa$ and torsion $\tau$ — from any space curve, build the Frenet-Serret frame, write the coupled ODE governing its evolution, and quote the fundamental theorem that these invariants determine the curve up to rigid motion. The next section steps up a dimension: surfaces have two principal curvatures at each point, and combining them gives the Gaussian and mean curvatures that classify local surface shape.

References

Garrity, T. (2002). All the Mathematics You Missed: But Need to Know for Graduate School. Cambridge University Press, ch. 7.
do Carmo, M. P. (2016). Differential Geometry of Curves and Surfaces (2nd ed.). Dover, ch. 1.
Pressley, A. (2010). Elementary Differential Geometry (2nd ed.). Springer, ch. 2-3.
Spivak, M. (1999). A Comprehensive Introduction to Differential Geometry (3rd ed., Vol. 2). Publish or Perish, ch. 1.
Misner, C. W., Thorne, K. S., Wheeler, J. A. (2017). Gravitation. Princeton University Press, ch. 9 (Frenet frames in relativistic contexts).