Euler Spiral

Curvature is how hard a curve bends: the reciprocal of the radius of the circle that hugs it at a point. A straight line has curvature $0$ , a circle of radius $R$ has curvature $1/R$ everywhere along it, and joining the two directly means jumping from one to the other with nothing in between. The Euler spiral is what fills the gap — the curve whose curvature grows in exact proportion to the distance travelled along it.

Pick a construction, then drag to pan and pinch or scroll to zoom.

Construction Arc length 4.0 Steps 720

Trace walks a point along the curve with the circle that matches its curvature at that moment — watching that circle shrink at a steady rate is the whole definition, made visible. The strip under the plot is $\kappa$ against $s$ : a straight line for the spiral itself, and a ramp against a step for the road transition. With the canvas focused, the arrow keys pan, + and − zoom and 0 refits. Expand fills the screen; Esc leaves it.

Curvature proportional to distance

The spiral is defined by the Fresnel integrals:

x(t) = \int_0^t \cos\left(\frac{\pi u^2}{2}\right) du

y(t) = \int_0^t \sin\left(\frac{\pi u^2}{2}\right) du

The integrand has modulus $1$ , so the curve is traced at unit speed and the parameter $t$ is arc length. That is what makes the rest fall out in one line: the tangent points along $(\cos \pi s^2/2,\ \sin \pi s^2/2)$ , so the tangent angle is $\varphi(s) = \pi s^2 / 2$ , and the curvature is its derivative,

\kappa(s) = \frac{d\varphi}{ds} = \pi s.

Straight at the origin, and bending harder the further you go, without limit. After $n$ full turns of the tangent the curve has covered an arc length of $2\sqrt n$ and its osculating circle is down to a radius of $1/2\pi\sqrt n$ , so the coils crowd in on themselves while the two branches wind onto the points $\pm(\tfrac12, \tfrac12)$ without ever reaching them. Euler described the curve in 1744; the limit points took him until 1781.

The same spiral turns up in optics, where it is usually named after Cornu. Arc length along it is distance across an aperture, each sample contributes a phase $\pi s^2/2$ picked up on the way to the screen, and the straight line joining two points of the spiral is the amplitude arriving from the strip of aperture between them. The tight coils at the ends are why the edge of a shadow is a set of fringes rather than a step.

Turning by hand

The curve can also be walked out, with no integral anywhere. Take unit steps, and before step $i$ turn by $i\theta$ :

after $i$ steps the heading is $\theta \cdot i(i+1)/2$ ,
and the distance covered is $i$ ,

so the turn per unit distance is about $\theta i$ — linear in arc length, which is the definition again. It is Euler’s method applied to $\varphi' = \kappa$ with a step of one: matching headings puts the walk at arc length $s = i\sqrt{\theta/\pi}$ on the curve above, so it is the Fresnel spiral drawn $\sqrt{\pi/\theta}$ times larger, converging on it as $\theta \to 0$ . This page used to carry a dozen lines of MATLAB doing exactly that, and a still of the result; the two sliders now run the same loop as you move them.

Turn $\theta$ up and the spiral stops being a spiral. Headings are only ever multiples of $\theta$ , so when $\theta$ is a rational multiple of $360°$ — say $\theta = 360°p/q$ — the heading sequence repeats with period $2q$ , because $i(i+1)/2$ advances by a multiple of $q$ over that many steps. A repeating sequence of headings means a repeating sequence of steps, so the walk copies one shape forever, and when that period’s steps cancel — which they do at every $\theta$ I have tried — the shape closes into a rosette. Every rational $\theta$ is therefore a resonance, but most of the periods are far too long to see: $\theta = 2°$ closes after 360 steps, $\theta = 2.01°$ after 24,000, and the slider stops at 2,400.

What it is for

A curve made of a straight and a circular arc has a step change in curvature at the join. For anything travelling along it at speed $v$ , lateral acceleration is $v^2\kappa$ , so a step in curvature is a step in sideways force — acquired instantaneously, in theory, and in practice by the vehicle sliding, the passengers lurching, and the track wearing at one spot.

Slot an Euler spiral in between and the curvature ramps instead of jumping. Over a transition of length $L$ into an arc of radius $R$ ,

\kappa(s) = \frac{s}{RL}, \quad \varphi(s) = \frac{s^2}{2RL}, \quad A = \sqrt{RL},

where $A$ is the clothoid parameter that fixes the scale, and the spiral turns through $\varphi(L) = L/2R$ before the arc takes over. At constant speed the lateral acceleration now climbs linearly with time, so its rate of change — the jerk $v^3/RL$ — is constant, and bounding that number is what actually sets $L$ in a design standard. On a railway the cant is ramped along the same length, for the same reason.

There is a geometric price, and the viewer draws it. Both curves in that mode run between the same two straights — which is how the choice actually presents itself, since the straights are fixed by where the road has to go and only the corner is up for negotiation — meeting at a deflection $\Delta$ . The plain curve of radius $R$ touches both straights. The transitioned one cannot: it has to hold the same arc clear of them by

p = \frac{L^2}{24R},

and start into the corner earlier by about $L/2$ , to leave room for the spirals. Those two gaps are the whole difference between the curves on screen, and the readout gives both, with the approximation above alongside the offset the geometry actually produces. Push the transition length to its maximum and the circular arc vanishes entirely: the corner becomes two spirals meeting nose to nose, each turning through $\Delta/2$ .

The curve is also the one a driver produces anyway — turning a steering wheel at a constant rate while holding speed traces a clothoid — which is why it fits the problem so exactly. Talbot worked it into railway practice in The Railway Transition Spiral, and the same shape is used for the vertical loops on roller coasters, where the point is to cap the peak acceleration rather than the jerk, and in font and vector-drawing tools, where a spline with piecewise-linear curvature is about the smoothest thing a curve can do.

Notes on the viewer

Curves are sampled by how far the tangent has turned, not by even steps of the parameter. The phase in the Fresnel integrals grows quadratically, so a step fine enough for the coils at $s = 8$ takes twenty-five times the samples per unit length that the straight middle needs, and one coarse enough for the middle cuts the coils into visible polygons; stepping so that the heading advances by a fiftieth of a radian each time spends the samples where the bending is. The integral itself is the midpoint rule, and lands within $2 \times 10^{-5}$ of tabulated values of $C(t)$ and $S(t)$ — four decimal places — for every range the slider offers.

Each sample carries its own signed curvature, in closed form, rather than having it differenced back out of the polyline afterwards — the strip plots that array directly. The transition alignment is assembled from three primitives (straight, arc, spiral), each advancing a running position, heading and arc length, so the pieces meet with matching tangents by construction rather than by fitting.

Rendering is plain canvas 2D. Pointer events cover mouse, trackpad and touch through the same code path: a two-finger pinch reads the change in spread between the pointers, and a flick leaves a short glide behind it. The centre of the view is clamped to stay over the curve, since one determined scroll otherwise leaves a blank canvas with no clue which way to come back.

Download the viewer source