# Measuring Curvature with a Bike

**Figure 1.**

*The center \(C\) of curvature of a bike’s rear track (see Figure 1) lies at the intersection point of the two axles’ extensions, i.e., of the normals to the front and rear tracks.*

I speak here of a mathematician’s bike, namely of a fixed length segment \(RF\) whose front \(F\) moves along a prescribed path and whose rear \(R\) has velocity vector constrained to the line \(RF\).

[1] posed the problem of finding a *geometrical* proof for this neat fact. I offer such a proof/explanation here, along with a few additional observations.

**Figure 2.**

**the claim for \(\alpha=\textrm{const}\).**

^{1}It remains to remove the constancy assumption, i.e., to explain why curvature \(\kappa\) does not in fact depend on the variation of \(\alpha\) but only on \(\alpha\) itself (and on the length \(l=RF\)).

**Figure 3.**Proving (1) by applying "

*ω=v/r*" to compute the angular velocity of

*RF*, i.e., the curvature at

*R*.

\[ \kappa = \frac{d\theta}{ds} = \frac{v \sin \alpha}{l} = \frac{\tan \alpha}{l}, \tag1 \]

proving the independence of \(\kappa\) on \(d\alpha/ds\) and thus justifying the original claim. Actually, the claim also follows directly from \((1)\), which yields \(\kappa ^{-1} = l \cot \alpha\) and coincides with \(RC = l \cot\alpha\) from Figure 1.

**Figure 4.**

*κ=KR*, where

*K*is the intersection point of the rear axle with the direction line of the front wheel. Here,

*l=RF=1*.

\[\kappa = KR,\]

assuming \(l=1\) for simplicity. Indeed, the triangles \(\Delta FRC\) and \(\Delta KRF\) are similar, so that (taking \(l=RF=1\)),

\[\frac{KR}{1} =\frac{1}{RC}, \quad \ \ \hbox{i.e.} \ \ \quad KR\cdot RC=1. \]

Thus, \(KR= RC ^{-1} = (\kappa ^{-1} ) ^{-1} = \kappa\), as claimed.

**Figure 5.**Measuring the curvature of the rear track with the bike light.

\[ \kappa=d/ \rho + O(d^3).\]

From now on I will probably always think of \(\kappa\) when riding a bike at night.

**Proofs which also explain**

^{1}*why*probably deserve a special name, something like “exproof.”

*The figures in this article were provided by the author.*

**References**

[1] Alexander, J.C. (1984). On the Motion of a Trailer-Truck. *SIAM Review, 26*(4), 579-580.