The curve that bends as far as it has gone

There are two ways to describe a road. One is a map: the coordinates of every point. The other is the instructions a driver needs: go straight for 200 metres, then begin turning, more and more sharply, until the turn is tight enough, then hold it. The second description has no coordinates in it at all. It says only how sharply to turn at each distance travelled, and yet it fixes the road completely, apart from where it starts and which way it points.

The length belongs to the journey argued that a curve’s length is a property of how it is travelled, not of the set of points it covers. The driver’s description goes further: once a curve is travelled at unit speed, so that the parameter is the length itself, one function of that length — the curvature — determines everything else. This essay draws curves from that one function, and follows the simplest non-trivial case, a curve whose bending grows in proportion to the distance gone. It is Euler’s spiral, and it turns up in railway engineering, in optics and in the theory of Gauss sums, under three different names; this essay follows the mathematics and leaves the engineering and the optics to their own subjects.

Turning at a stated rate #

Travel a plane curve at unit speed and write $\theta(s)$ for the direction of travel after a distance $s$. The curvature is how fast that direction turns, $\kappa(s) = \theta'(s)$. A circle of radius $r$ has constant curvature $1/r$; a straight line has curvature 0. The turning number of a closed curve is the total of its curvature divided by a full turn.

Given $\kappa$ as a function of $s$, the rest follows by integration twice over. The direction is $\theta(s) = \theta_0 + \int_0^s \kappa$, and since the curve moves one unit per unit of $s$ in that direction,

$$x(s) = x_0 + \int_0^s \cos\theta(t)\,dt, \qquad y(s) = y_0 + \int_0^s \sin\theta(t)\,dt .$$

The three constants $x_0$, $y_0$ and $\theta_0$ are the starting point and starting direction, and changing them moves the whole curve rigidly. So the curvature as a function of arc length — the curve’s natural equation — determines its shape exactly, with nothing left over. This is the plane version of a theorem whose space version needs a second function, the torsion, to say how the curve twists out of its plane.

Constant curvature gives a circle, as it must. Curvature $1/s$, falling as the length grows, gives the equiangular spiral of a nautilus shell, whose turns widen in proportion and which crosses every ray from its centre at the same angle. A curvature that oscillates as a cosine of the length gives a curve whose direction oscillates as a sine, and that is the sine-generated curve that Walter Langbein and Luna Leopold proposed in 1966 as the shape of a river meander. They argued that among all ways of bending a fixed length of channel between two points, it minimises the total squared change of direction, and rivers on flat floodplains match it closely. In each case the curve was drawn by integrating its curvature with small steps and nothing else; no formula for the shape was used.

Bending in proportion to distance #

The simplest curvature after a constant is the length itself: $\kappa(s) = s$. Then $\theta = s^2/2$ and

$$x(s) = \int_0^s \cos\frac{t^2}{2}\,dt, \qquad y(s) = \int_0^s \sin\frac{t^2}{2}\,dt .$$

At $s = 0$ the curve is momentarily straight. Going forward it bends to the left more and more sharply, and going back it bends to the right; the two arms are images of each other under a half-turn about the origin. Because the curvature keeps growing, each arm turns through larger and larger angles in shorter and shorter distances and spirals tightly round a point.

The two integrals are Fresnel’s integrals, and their values at infinity are exactly known:

$$\int_0^\infty \cos\frac{t^2}{2}\,dt = \int_0^\infty \sin\frac{t^2}{2}\,dt = \frac{\sqrt{\pi}}{2} \approx 0.8862 .$$

The derivation is the same rotation of a contour that evaluates the Gaussian integral, which is why $\sqrt\pi$ appears here and in the constant under every bell curve. The arms wind into $\pm(\sqrt\pi/2, \sqrt\pi/2)$, the red dots of the hero. Euler met the curve in 1744 in his work on the bending of elastic strips and computed these limits decades later, in 1781.

How close does the arm get? At length $s$ the curve is turning with radius $1/s$, so it is circling its limit at about that distance, and the computed distances confirm it: 0.1000 at $s = 10$, 0.03333 at 30, 0.01111 at 90. Each arm is infinitely long and lies within a bounded region, so its later part consists of infinitely many tighter and tighter turns of total length infinity in a shrinking disc. That combination — infinite length, finite reach — is exactly what which curves have a length at all separated from the behaviour of a well-behaved curve on a closed interval: restricted to any finite stretch the spiral has a perfectly finite length, and only the whole arm is unbounded.

Two spirals, two kinds of infinity #

The equiangular spiral of the second figure makes an instructive contrast. Followed inwards, it too makes infinitely many turns round a point, and its turns shrink geometrically. But its inward arm has finite length: each turn is a fixed fraction of the one outside it, so the lengths of the turns form a geometric series, and Evangelista Torricelli summed it in 1645, decades before calculus. In the natural equation the difference is plain. The equiangular spiral has curvature $1/s$, measured from the centre, so the length remaining to the centre is $s$ itself, finite at every point. Euler’s spiral has curvature $s$, and there is no point at which the remaining length is finite.

So two curves that look alike in a drawing — tight turns crowding round a point — can differ completely in length, and the natural equation says which is which at a glance: in one the curvature blows up at a finite length, in the other only as the length runs to infinity. The same lesson, that a picture does not show length, is the one the staircase taught with corners rather than turns.

What the total of the bending counts #

Add up the curvature along a closed curve and the total is a whole number of full turns, since the curve ends pointing the way it started; the number is the turning number. For a curve in space there is no turning number, but the total of the curvature still means something: Fenchel proved that it is at least a full turn for any closed curve, and Fáry and Milnor that it exceeds two full turns for any knotted one. Euler’s spiral has infinite total curvature on each arm, $\int_0^\infty s\,ds$, which is the analytic form of its infinitely many turns.

The natural equation also explains why a sine wave is a circle seen from the side in a sense that matters here. The meander’s direction oscillates as a sine of the length; the sine wave’s height oscillates as a sine of horizontal distance. They are different curves, and confusing them is the commonest error in drawing a river: a sine wave of large amplitude has nearly vertical sections and sharp tops, while a meander of the same reach turns evenly all the way round, because what is sinusoidal in it is the turning.

Curvature that repeats #

A curvature that is periodic in length makes a curve built of identical pieces, and the natural equation decides at once what the pieces add up to. Take the curvature $a + b\cos s$, which repeats every $2\pi$ of length. Over one period the tangent turns by the integral of the curvature, $2\pi a$, since the cosine contributes nothing. So the second period of the curve is the first one turned through the angle $2\pi a$, the third is the second turned through the same angle, and so on: each period is the previous one moved by a fixed rigid motion.

A rigid motion that turns through an angle which is not a whole number of revolutions is a rotation about some centre. If $a$ is a fraction $p/q$, then $q$ periods turn through $p$ whole revolutions, the motion repeated $q$ times is the identity, and the curve closes: with $a = 2/5$ and $b = 2$ it closes after five periods into the rosette in the figure, its end landing on its start to within $10^{-6}$. If $a$ is irrational the rotations never return, and the curve winds round the centre for ever, filling a ring densely, as $a = 1/\sqrt 7$ does after forty periods. It is the familiar split between rotations by rational and irrational fractions of a turn, visited on a whole curve rather than on a point.

When $a$ is a whole number, the turning over a period is a whole number of revolutions, and the motion is not a rotation but a translation: each period is the last one slid along. The step is the integral of $e^{i\theta(s)}$ over one period with $\theta = as + b\sin s$, and for $a = 1$ that integral is a classical one: by the expansion of $e^{ib\sin s}$ in Bessel functions, it equals $2\pi$ times $J_1(b)$, up to sign. At $b = 2$ the computed step is 3.62367, matching $2\pi J_1(2)$, and the curve drifts along a line. But $J_1$ has zeros, the first at $b = 3.831706$, and there the step vanishes: the curve with curvature $1 + 3.8317\cos s$ closes after a single period, ending exactly where it began. Whether a curve closes is a question about its curvature alone, and here its answer is a zero of a Bessel function.

Measured across a range of $b$, the step per period traces $2\pi|J_1(b)|$ to four decimal places: it rises from nought, peaks near 3.6 at $b \approx 1.8$, and falls to nought at 3.8317, then rises and falls again with smaller and smaller arches, vanishing at 7.0156 and 10.1735 below eleven. At each zero the curve closes after one period, and each closed curve has one more small loop than the last, because a larger $b$ makes the curvature swing further negative and the curve double back on itself more often within the period. Between the zeros the curve drifts, and it drifts backwards wherever $J_1(b)$ is negative. The zeros themselves are the same numbers that give the radii of the dark rings in the pattern of light through a circular hole and the nodes of a vibrating drum, which is the kind of coincidence a natural equation keeps producing: what the curve does over one period is an integral of $e^{i\theta}$, and integrals of that shape are where Bessel functions come from.

The spiral taken in steps #

Replace the integral by a sum. Take unit steps and after each one turn by an angle that grows in proportion to the step number, so that the direction after $k$ steps is $a k^2/2$. For a small turning rate $a$ this is Euler’s spiral drawn as a polygon, as the first walk in the next figure shows.

With larger turning rates the same rule produces something stranger: the walk spirals, unwinds, travels, and spirals again, in a pattern Michael Berry and Jonathan Goldberg called curlicues in 1988 and analysed by renormalisation. When $a/2\pi$ is a fraction, the turning angles repeat and the walk is built from copies of one finite piece. With $a = 4\pi/7$ each step points along a seventh root of unity indexed by a square, and every seven steps the walk adds the sum

$$\sum_{k=0}^{6} e^{2\pi i k^2/7},$$

which is the quadratic Gauss sum. Its length is exactly $\sqrt 7$, as one sum squared two ways proved for every prime, so the walk advances by $\sqrt 7$ each period and the figure is repeated along a line.

This is the surprising connection. The curve that engineers know as the transition spiral and opticians as Cornu’s spiral is the continuous limit of the sums that Gauss used to prove the law of quadratic reciprocity. All three are sums or integrals whose phase grows as a square. The same quadratic phase drives the slope a Gauss sum leaves behind, where Riemann’s nowhere-quite-differentiable function is a sum over squares.

Why a square, every time #

The quadratic phase is not a coincidence of these three examples. When phase grows linearly with position the terms just rotate uniformly, and its sum or integral is a circle. The first correction to linear growth, for any smooth phase, is quadratic — that is Taylor’s theorem — so a quadratic phase is what appears near any point where the linear part vanishes. In optics that point is the straight line to the observer, where the path length is stationary; in the railway it is the start of the transition, where curvature is nought; in the Gauss sum it is the multiplicative structure of squares modulo a prime. The method of stationary phase, which approximates any oscillatory integral near such points, reduces each case to a Fresnel integral, and so to a piece of Euler’s spiral.

A spiral only as good as its integration #

The figures are numerical integrations with small steps, and a curve drawn from its curvature is only as accurate as the integration; for the spiral’s far arms the steps were refined until the distance to the limit point agreed with $1/s$ to within a few per cent, and the integration of curvature was checked against the exact Fresnel form to $10^{-4}$. No figure can show the infinitely many turns at the end of an arm; the drawing stops at $s = 7$.

The meander figure shows the shape Langbein and Leopold proposed, not any actual river, and the claim that rivers follow it is quoted, not tested here. The closing curves are found by integrating over one period, and a curve that closes only after many periods, or one with a rational $a$ of large denominator, would need longer integration than the figures use; the irrational case is shown after forty periods, which cannot distinguish an irrational $a$ from a fraction with a denominator above forty. And the curlicue walk is shown at three turning rates only; its behaviour for a typical irrational rate is a subject in itself.

Curves integrated, Euler and Bessel quoted #

Computed here: every curve from its curvature by midpoint integration; the spiral checked against the direct Fresnel form, and a constant curvature of 1 checked to close a circle; the distances to the limit point for $s$ from about 1.4 to 100; the curves with curvature $a + b\cos s$, closing after five periods at $a = 2/5$ to within $10^{-6}$, and the step per period at $a = 1$ against $2\pi|J_1(b)|$ at 45 values of $b$, with the zeros 3.8317, 7.0156 and 10.1735 found by bisection; the walks, and the length $\sqrt 7$ of the seven-step Gauss sum.

Quoted: Euler’s work of 1744 and 1781, the spiral’s uses in railway design and in optics, the expansion of $e^{ib\sin s}$ in Bessel functions, Langbein and Leopold’s meanders, and Berry and Goldberg’s curlicues.

Still open: why a river chooses the curve #

The sine-generated curve fits real meanders well, and Langbein and Leopold derived it from a principle of minimum total bending. Whether that principle is the actual cause is not settled. Rivers are shaped by flow, erosion on the outside of bends, deposition on the inside, and the occasional cutting-off of a loop, and models of these processes produce meanders that are skewed, asymmetric and growing in ways a pure sine-generated curve is not. Which mechanism selects the shape, and how much of the fit is a consequence of any smooth curve with oscillating curvature looking like a meander, is studied and disputed in geomorphology. The mathematics here is the easy half: once the curvature is known, the curve is fixed.

There is a mathematical question on the other side. For a typical irrational turning rate, how far does the curlicue walk of $N$ steps get from its start? Hardy and Littlewood showed in 1914 that for rates whose continued fractions have bounded quotients the distance is at most about $\sqrt N$, and the finer statistics — the distribution of the distance for a random rate — have been worked out only in this century, through a connection with the geometry of the modular surface. The picture of a random curlicue is still not described by any theorem as simple as the one for Euler’s spiral.

A curve fixed by how it bends #

A plane curve travelled at unit speed is determined, up to a rigid motion, by its curvature as a function of length. Curvature equal to length gives Euler’s spiral, whose arms are infinitely long and wind into points $\sqrt\pi/2$ along each axis, approaching at distance $1/s$. A curvature that repeats builds a curve from identical pieces, which closes when the turning per period is a rational fraction of a revolution and, at a whole revolution, only at the zeros of a Bessel function; and Euler’s spiral, summed in unit steps with a turning angle growing as a square, becomes the Gauss sum.