The turns a tangent makes

Draw a closed curve on paper without lifting the pen, and without ever stopping or turning a sharp corner — the curve may cross itself as often as it likes, but at every moment it is moving in some definite direction. Follow that direction once round the curve and it comes back to where it started, having turned some whole number of times. That number is the curve’s turning number.

A circle traced anticlockwise turns once. A figure-eight turns once anticlockwise round one lobe and once clockwise round the other, nought in all. A limaçon with a small loop inside turns twice — once for the outer sweep and once more for the loop — and every further inward loop adds one: the next two curves turn three and four times. A curve can also turn negatively, and the three-lobed star, whose lobes are swept clockwise while the curve as a whole runs anticlockwise, turns −2 times.

The theorem this essay is about is that the turning number is the only thing that matters. Hassler Whitney proved in 1937, from a suggestion of William Graustein’s, that two such curves can be deformed into each other — through curves that also never stop and never form a corner — exactly when their turning numbers are equal. Earlier essays here classified loops in a space up to deformation, by the group they form. This one classifies something stranger: curves in the plane, where every loop can be shrunk, up to the deformations that keep the curve moving.

A loop in the circle of directions #

The turning number is a fundamental-group class in disguise.

At each moment the curve’s direction of motion is a point on the circle of directions. As the curve is traced once round, that point traces a closed loop in the circle — and loops in a circle are classified by a single whole number, how many times they go round. Unroll the angle, never resetting it at 360°, and for the limaçon it climbs through exactly 720°: two turns. It climbs fastest through the inner loop, where the curve bends most sharply, and it never falls, because the limaçon always bends to the left.

So the turning number is the winding number of the velocity about nought — the same invariant as the winding of a path round a removed point, applied to the velocity instead of the position. For a curve given by a formula it can be read off directly: the limaçon $e^{it} + 0.8\,e^{2it}$ has velocity $i e^{it}(1 + 1.6\,e^{it})$, which winds once for the first factor and once more for the second, because $1.6 > 1$. The curve with three inner loops, $e^{it} + 0.4\,e^{4it}$, has velocity $i e^{it}(1 + 1.6\,e^{3it})$, and winds $1 + 3 = 4$ times. The three-lobed star, $e^{it} + 0.75\,e^{-2it}$, has velocity $i e^{it}(1 - 1.5\,e^{-3it})$, and winds $1 - 3 = -2$ times.

Why the number cannot change, and why it is enough #

One direction of Whitney and Graustein’s theorem is easy. Deform a curve continuously, keeping its velocity non-zero, and the direction of motion moves continuously too; so its loop in the circle of directions is deformed continuously, and the number of times it goes round cannot jump. Curves with different turning numbers cannot be deformed into one another without, at some moment, stopping.

The limaçons show it happening. In the family $e^{it} + a\,e^{2it}$ the velocity is $i e^{it}(1 + 2a\,e^{it})$, which vanishes exactly when $2a = 1$ and $e^{it} = -1$. For $a$ below a half the curve is a dimpled oval turning once; for $a$ above a half it has an inner loop and turns twice; and at $a = \tfrac12$ exactly it is the cardioid, the heart-shaped curve with a single cusp, where the inner loop has shrunk to a point and the curve, for one instant, stops. Shrinking a loop away is not a way of avoiding the theorem; it is the clearest example of it. A loop can be made as small as one likes, and its contribution to the turning number stays a full turn until the moment the curve stops to let it go.

The other direction is the theorem’s content: equal turning numbers are enough.

A circle traced twice and a limaçon with one inner loop look nothing alike, but both turn twice. The family $s\,e^{it} + (1 - s + 0.8s)\,e^{2it}$ moves one into the other: at $s = 0$ it is the doubled circle, at $s = 1$ the limaçon, and in between the inner loop grows out of the second circuit. Along the whole deformation no curve’s speed falls below 0.6, so the deformation never stops, and every curve along it turns exactly twice.

Whitney’s proof constructs such a deformation in general. Reparametrise both curves so that their directions of motion turn at matching rates; then average the two curves, adjusting the average so that its velocity never vanishes. The averaging works because two velocity loops in the circle with the same winding number can be deformed into one another, and the curve can be rebuilt from its velocity by integration, with a correction to make it close up. Every step is explicit; the cleverness is in the correction.

The cusp a figure-eight costs #

A circle and a figure-eight have turning numbers 1 and 0, so no such deformation joins them. Any deformation at all must therefore pass through a curve that stops somewhere.

The family $x = \cos t$, $y = (1 - s)\sin t + \tfrac12 s \sin 2t$ is a circle at $s = 0$ and a figure-eight at $s = 1$. Its velocity at the left-hand end, $t = \pi$, is $(0,\, -(1 - s) + s)$, which vanishes at exactly $s = \tfrac12$. There the curve has a cusp: it comes into the point, stops, and leaves in the direction it came from. Before the cusp the curve turns once; after it, nought times. The pinched end has turned itself inside out, and the only way through was to stop.

That moment is exactly the kind of singularity Whitney studied elsewhere, in maps of the plane that fold it over itself: generic families of curves acquire cusps at isolated moments, as generic maps of the plane acquire folds and cusps along curves and at points. A count that is preserved by everything smooth changes only where something stops being smooth.

Reading the number off the crossings #

Following the tangent all the way round is not the only way to find the turning number. Whitney found a formula that reads it from the crossings.

Start at the topmost point of the curve, which is on its outer boundary, and let $\mu$ be $+1$ if the curve leaves it leftwards (anticlockwise round the outside) and $-1$ otherwise. Travel round the curve. Each crossing is passed twice; give it the sign $+1$ if, on the second passage, the curve cuts across its first passage from right to left as seen along the first passage, and $-1$ the other way. Whitney’s formula of 1937 is

$$\text{turning number} = \mu - \sum_{\text{crossings}} \varepsilon .$$

The first curve in the figure leaves its top leftwards, so $\mu = +1$, and its two crossings have opposite signs, so the turning number is 1. The third has $\mu = -1$ and crossings $+, -, +$, so its turning number is $-1 - 1 = -2$. On all 300 random curves tried — each a sum of six terms $c_k e^{ikt}$ with random coefficients, $k$ from $-3$ to 3 — the formula agrees with the turning number computed from the derivative, on every one.

Why the formula holds is easiest to see by taking the curve apart. At each crossing, cut the two strands and reconnect them the other way round, keeping the direction of travel — the curve’s orientation decides which of the two reconnections does that. Doing it at a crossing changes nothing about the tangent except within a tiny neighbourhood, where the two corners it creates turn through equal and opposite angles, so the total turning is unchanged. Do it at every crossing and the curve falls apart into a collection of disjoint simple closed curves, each turning $+1$ or $-1$ by Hopf’s theorem below. The turning number is the number of anticlockwise pieces minus the number of clockwise ones. Whitney’s signs keep track of how each smoothing splits or merges pieces, and the start at the outermost point fixes the orientation of the outer piece, which is where $\mu$ comes from.

The same total can be computed in a third way, by curvature. The rate at which the direction of motion turns, per unit of distance travelled, is the curve’s curvature, so the turning number is the total curvature divided by $2\pi$. For a polygon the turning happens all at once at the corners, and the turning number is the sum of the exterior angles divided by $2\pi$ — the fact that a simple polygon’s exterior angles add up to 360° is Hopf’s theorem for polygons. In space the direction of motion is a loop on a sphere rather than a circle, the total curvature no longer has to be a multiple of $2\pi$, and the analogous statements become inequalities: a closed curve in space turns through at least $2\pi$, and a knotted one through more than $4\pi$.

Crossings and turns differ in parity #

Whitney’s formula has an immediate consequence that a drawing can check at a glance.

The turning number is $\pm 1$ plus a sum of $\pm 1$’s, one per crossing, so it has the opposite parity to the number of crossings. Of 400 random curves, not one lands where the two numbers have the same parity: 83 curves with no crossings turn $\pm 1$; 124 with one crossing turn 0 or $\pm 2$; 94 with two crossings turn $\pm 1$; and so on, up to two curves with seven crossings, each turning nought.

Two consequences are worth separating out. A curve with no crossings at all — a simple closed curve, the kind the Jordan curve theorem divides the plane with — must turn exactly once, one way or the other. That is Heinz Hopf’s Umlaufsatz of 1935, the theorem of turning tangents, and every driver who has gone once round a roundabout knows it: the car ends pointing the way it started, having turned through exactly 360°. And a curve with one crossing — a figure-eight, or a limaçon with a loop — can turn 0 or ±2 times, but never once. The fact that a figure-eight cannot be deformed into a circle without stopping is visible in a single crossing.

The direction a contour is drawn in #

The sign of a simple curve’s turning number has a practical use wherever shapes are described by their outlines. A letter in a font, a country on a map, a part on a cutting machine is stored as a list of closed contours, and a program filling them in needs to know which contours are outer boundaries and which are holes — the inside of an o, a lake within a country. The usual convention is that outer boundaries run one way and holes the other, and the program decides which way a contour runs by computing its signed area, walking once round it and adding up. For a simple contour the sign of that area and the sign of its turning number agree: anticlockwise contours have positive area and turn $+1$.

For a contour that crosses itself the two numbers come apart. A figure-eight has turning number nought, and its signed area is the difference of its two lobes’ areas, which can be anything. That is why fill rules for self-crossing outlines are stated in terms of the winding number of the outline round each point — how many times it goes round that point — rather than in terms of the turning of the outline itself: the two lobes of a figure-eight are wound round once in opposite directions, $+1$ and $-1$, and both are filled, although the outline as a whole does not turn at all.

What the theorem leaves out, and what generalises #

The turning number classifies closed curves up to deformations that keep them moving — regular homotopies. It says nothing about deformations that also keep the curve from crossing itself, which is a much more restrictive kind of motion; for simple closed curves that question has a short answer, since every one is a circle in disguise and can be moved to a round circle without ever crossing itself, turning +1 or −1 according to the direction it is traced.

The theorem’s real legacy is in higher dimensions. Stephen Smale extended Whitney and Graustein’s classification in 1957 to spheres moving in space without creases, and found that the relevant invariant for a sphere in three-dimensional space lies in a group that turns out to have only one element. So a sphere can be turned inside out — its outer surface becoming its inner surface — by a motion that lets it pass through itself but never creases it. The circle has no such motion: a circle traced anticlockwise turns $+1$ and traced clockwise $-1$, and no regular homotopy joins them. That a sphere can be everted when a circle cannot is one of the most surprising results of twentieth-century topology, and it is exactly the circle’s turning number, generalised and found to vanish.

There is a similar invariant on surfaces other than the plane, where the turning number has to be measured against a field of reference directions, and on a surface where no field of directions can be combed flat — the sphere — it is defined only up to an ambiguity. On the sphere, curves are classified by turning number modulo two, and every closed curve on the sphere can be deformed into one of exactly two.

What the computations show and what they assume #

The turning numbers are computed from the derivative, sampled at two thousand points and summed as angle changes; the sum lands within a millionth of a whole number on every curve here. The crossings are found by intersecting the segments of a nine-hundred-point polygon through the curve, which can miss a crossing inside a loop smaller than the spacing; the random curves are drawn from a family whose loops are not that small, and curves whose speed falls below six per cent of its maximum — the near-cusps where a tiny loop could hide — are excluded from the sample. Within that sample Whitney’s formula holds on every curve, and so does the parity, as they must, since both are theorems; the sample’s role is to check the computation, not the mathematics.

Still open: counting the curves of each kind #

How many essentially different closed curves have $n$ crossings? Whitney and Graustein’s theorem says which curves can be deformed into one another when the crossings are allowed to change; it says nothing about how many combinatorially different drawings there are with a fixed number of crossings, where the deformation is not allowed to create or destroy crossings. Those are plane curves in the sense of Vladimir Arnold, who introduced three further invariants for them in 1994, and their enumeration by number of crossings has been carried out by computer to moderate sizes, with no formula known and the growth rate of the counts known only approximately. The turning number is the coarsest of a family of invariants that becomes richer the more strictly the curves are held still.

One number for every curve that moves #

The direction of motion of a closed curve is a loop in the circle of directions, and the number of times it turns is the curve’s turning number — 1 for a circle, 0 for a figure-eight, one more for each inward loop. Whitney and Graustein proved that this number decides exactly when one such curve can be deformed into another without stopping; Whitney’s formula reads it from the crossings, so its parity is opposite to the number of crossings, and a simple closed curve turns exactly once. Between a circle and a figure-eight every deformation must stop at a cusp, which is the price of changing the number and the reason a circle, unlike a sphere, cannot be turned inside out.