Roads, rails, roller coasters and light · one curve

The Bend That Doesn’t Jolt

Drive from a straight road directly onto a circular bend and the sideways push arrives all at once. Ask instead for it to build at a steady rate and the road is forced into one particular curve, whose curvature grows in proportion to its length: the Euler spiral, or clothoid. The mathematics of springs, the physics of light and the engineers of railways and roller coasters each found it without knowing about the others. Drive it and watch the ornament on the mirror, ride a loop that escapes the circle’s unavoidable 6 g, and read the fringes of laser light past a razor blade off the same curve, checked against the table Fresnel computed by hand.

Picture a road that runs dead straight and then, at a painted line, becomes an arc of a circle. Nothing looks wrong on the map. Drive it, though, and at that line something happens all at once. On the straight nothing pushes you sideways. On the circle something does, steadily. And between the two there is no between.

The sideways push on a curve depends on two things only: how fast you are going and how sharply the road bends. Call the sharpness the curvature, κ, which is one over the radius. Then

sideways acceleration a = v²κ
its rate of change, the jerk = v³ · dκ/dsv = speed (held constant), s = distance along the road

On a straight that runs into a circle, κ jumps from zero to 1/R across a line of paint, so the jerk is not merely large: it is unbounded. Everything loose in the car, including your head, gets the whole change of force in no time at all.

So ask for the gentlest possible alternative: let the push build up at a steady rate. Constant speed, constant jerk. Read the second line of the box backwards and the answer is forced: dκ/ds must be constant, so the curvature has to grow in exact proportion to the distance travelled. That one sentence defines a curve. It is called the Euler spiral, or the clothoid, or the Cornu spiral, depending on which of the several fields that found it you learned it from.

1. The curve that asks nothing else

Here it is. It starts perfectly straight in the middle, and every step outward bends it a little more than the step before, so it winds tighter and tighter into two “eyes” that it circles for ever without reaching. Slide the point out along it. The dashed circle is the circle the curve is momentarily following, and its radius shrinks exactly as fast as the length grows.

Drawing it takes one piece of integration nobody can do in closed form. If the heading angle after distance s is θ = s²/2A² (that is what “curvature grows in proportion to length” integrates to), then the position is the running total of cos θ and sin θ. Those two running totals are the Fresnel integrals, C and S. The engine on this page computes them by a power series near the middle and by an asymptotic formula out in the coils, and the verifier holds both against 30-digit values from an independent library and against brute-force numerical integration.

The number A sets the scale. A railway transition that takes a straight line up to a circle of radius R over a length L is simply the first stretch of this curve with A² = R·L. The verifier builds a whole bend (straight, spiral, arc, spiral, straight), integrates it step by step, and finds its spiral lying on the Euler spiral to within a hundredth of a micrometre.

2. Drive the bend

Below is a 60° bend on flat ground. The slider marked spiral sets how long the transition is: at zero the straight runs directly into the circle, the way railways were first laid. Hanging from the mirror is an ornament on a 25 cm string, which is a pendulum, and the most honest accelerometer a car has. Press drive.

With no spiral, watch the ornament. It does not simply lean over to its new angle. It is flung out to twice that angle and then swings back and forth about it, because at the paint line it was hanging straight down, already a full resting-angle away from where it now belongs, and nothing had eased it over. That factor of two is exact for an undamped pendulum hit by a sudden steady push, and it survives the small-angle approximation: in the car’s frame the pendulum simply finds itself released from rest in a tilted gravity, and a pendulum released from rest swings equally far to the other side.

Now give the bend a spiral and the swing that is left over shrinks. It shrinks by a precise rule: if the spiral takes T seconds and the ornament’s own swing is ω radians per second, what remains is

left-over swing = resting angle × |sin(ωT/2) ÷ (ωT/2)|undamped, small angles; the verifier integrates the pendulum and matches this within 0.5% of the resting angle

and that has a surprise in it. When the spiral lasts exactly one swing of the ornament, the formula gives zero: the ornament leans over and arrives at its new angle perfectly still. (Press the second button.) It is the same trick a crane operator uses to move a hanging load without leaving it swinging. The sweet spot depends on the length of the string, so no real road can hit it for every passenger, but the envelope is the point: past the first zero the left-over swing can never again exceed 22% of the resting angle, whatever the timing.

What the model is: a car at constant speed on level ground and a small-angle, undamped pendulum. Real railways and roads also tilt the track (cant, or superelevation), and the transition ramps the tilt up along with the curvature, so what a passenger feels is the part of the sideways push the tilt does not cancel. The shape of the argument does not change; the numbers do. In the small-angle model the resting angle is a/g in radians (14.6° at the default settings, where the exact angle is 14.3°).

There is one more thing a spiral costs, and surveyors have a name for it. A curve that tightens gradually cannot reach the circle at the same place a sudden one would: the circle has to sit a little further in from the straight to leave room, by an amount called the shift, very nearly L²/24R. For a 25 m spiral into a 250 m circle that is 104 mm, and the verifier measures it off the integrated track. It is why a transition cannot simply be painted onto an existing curve: the circle itself has to move.

3. The loop that tries to break your neck

Turn the bend on its side and it becomes a roller-coaster loop, and there the jolt stops being a matter of comfort. Take a perfectly circular loop of radius r. At the top the train must be going fast enough not to fall off; say the rider feels pressed into the seat at g_top there. Energy conservation fixes the speed at the bottom, and the circle fixes the sideways (here, upward) acceleration. Put them together:

felt at the bottom of a circular loop = g_top + 6 gfor any radius: a bigger circle needs more speed, and the two cancel exactly

Six g on entry, at the very least, whatever the size of the loop, and it arrives in an instant, because the train comes off level track (1 g) straight onto the circle. Now let the loop’s curvature grow with length instead. The slider below shares the curvature out between “constant” (a circle) and “growing in proportion to length” (a clothoid that starts from straight track), keeping the height and the feeling at the top the same.

The clothoid loop is tall and narrow, a teardrop, with a tight top and wide sides. It starts at 1 g because its curvature starts at zero, so there is no jolt at all on entry; its hardest moment comes partway up, and at the default settings it is about 3.8 g against the circle’s 6.5. It also needs less speed at the bottom to get round. The verifier checks the felt g a second way, by timing a rider round the curve and differentiating its position twice, without using the v²κ formula anywhere.

What the model is: a point rider on a frictionless track in a vertical plane. A real train has length, so different cars take the loop at different speeds; real track loses energy to friction and air; real loops are drawn from several pieces rather than one formula, and they are offset sideways so the exit can pass the entrance. None of that rescues the circle: the +6 g is a floor set by energy and geometry alone.

The first loops were circles

The Flip Flap Railway opened at Sea Lion Park on Coney Island in 1895 with a loop about 25 feet (7.6 m) across. It is often said to have given its riders 12 g. That figure traces back to a physics textbook that gives no working for it, and nothing could have measured it: as the coaster historian Nick Weisenberger puts it, “When Flip Flap Railway was operating in 1895, no reliable accelerometer technology existed yet.” His own estimate, from an assumed drop and speed, is 6 to 8 g. The arithmetic above says what can be said for certain: a circle gives at least 6 g on entry if the riders are to be in their seats at the top, and even if the car crawled over the top with the riders hanging in their restraints (−1 g there) it would still be 5 g at the bottom, a case the physicist Ann-Marie Pendrill works through in her 2005 paper on loop shapes.

The clothoid loop, Pendrill writes, was “first introduced by Werner Stengel”, and her reference to his firm’s history page places its first use on Revolution at Six Flags Magic Mountain, built by Anton Schwarzkopf and opened on 8 May 1976. Stengel’s engineering firm, which says it is behind more than 800 roller coasters, describes it this way: “By ingeniously incorporating the clothoid shape, we took the world by storm with the safely rideable looping.” For scale, Pendrill’s figures: “Children’s roller coasters may be limited to 2g, family rides often reach 3g, sometimes more, whereas many of today’s large roller coasters exceed 4g.”

4. The same curve, in light

Nothing about the curve is special to vehicles. It turns up wherever something adds up little contributions whose angle grows with the square of a distance, and light going past an edge does exactly that.

Shine a laser past a razor blade onto a wall. Geometry says the wall shows a sharp shadow. It does not. Every point of the wavefront that gets past the blade sends out its own little wave, and the wave from a point a distance u along the wavefront arrives late by an amount proportional to u². Add those little arrows up, each turned by an angle that grows as the square of its distance, and you are drawing the Euler spiral: the arrows lie head to tail along it. The light arriving at a point on the wall is the straight line (the chord) from where the blade cuts the spiral to the far eye, and the brightness is the square of that chord.

Three things fall out, and none of them is what geometry predicts. Light gets into the shadow, fading smoothly rather than stopping. Exactly on the edge of the geometric shadow the brightness is one quarter of the full beam: the chord from the spiral’s centre to one eye is half the chord from eye to eye, and brightness goes as the square. And just outside the shadow the wall is brighter than the unobstructed beam, by 37% at the first fringe, because there the chord from the cut to the far eye is longer than the straight line from eye to eye.

These numbers are checked two ways. The Cornu construction is compared with 30-digit values of the Fresnel integrals, and, independently, with a brute-force sum of 432,000 separate wavelets per point on the wall, using exact distances rather than the square-law approximation. The two agree to within a ten-thousandth of the full beam. And both agree with the table Fresnel himself worked out by hand around 1818 (below).

5. Found four times

The curve has been discovered at least three times by people who did not know it had been found before, and named each time after whoever found it. Nearly everything in this section comes from Raph Levien’s history of the curve (2008), checked where possible against the primary text.

• 1694A spring. Jakob (James) Bernoulli, in one sentence of a list in the Acta Eruditorum, poses the problem “To find the curvature a lamina should have in order to be straightened out horizontally by a weight at one end” (Levien’s translation). The weight bends each point of the strip in proportion to its distance from the end, so the pre-curved strip must have curvature proportional to length. Bernoulli wrote down the equation but, Levien notes, “did not draw its true shape”.
• 1744Euler. In an appendix to his Methodus inveniendi Euler characterises the curve (“rs = aa”: radius times length is constant), sees that it winds for ever round a point “which seems very difficult to find”, and leaves finding it as a challenge that “does not seem unworthy for geometers to exercise their strength”.
• 1781Euler again. Thirty-seven years later he finds the point himself, “by a happy chance and in an exceedingly peculiar manner”. In the units of the drawing above it is the eye at (½, ½), which the verifier confirms the engine’s spiral winds into.
• 1818Light. Augustin Fresnel, working on diffraction, derives the same integrals with no knowledge of Euler’s, and tabulates them to four decimals. His memoir computes the fringes beside a shadow by hand (the table is below).
• 1874The picture. Alfred Cornu plots the spiral accurately and turns it into a graphical calculator: the intensity is the squared distance between two points on the curve, which is the construction the razor-blade instrument above draws. At Cornu’s funeral in 1902 Henri Poincaré said he had “quickly replaced an unpleasant multitude of hairy integral formulas with a single harmonious figure, that the eye follows with pleasure” (Levien’s translation).
• c.1886The name. Ernesto Cesàro calls it the clothoïde, after Clotho, the Fate who spins the thread of life.
• 1890–91Railways. Railway engineers had been easing curves since Gravatt’s sine-curve transition of about 1828, William Froude’s proposal of about 1842 and Ellis Holbrook’s “true transition curve” of 1880. Arthur N. Talbot of the University of Illinois works out the spiral’s formulas in full as “a curve whose degree-of-curve increases directly as the distance along the curve”, first in the student journal Technograph, No. 5, 1890–91, then as a field book in 1899. Nobody connects his railway spiral with the clothoid in print until a book by Arthur Higgins in 1922.
• 1976Loops. Werner Stengel’s clothoid loop on Revolution.

Talbot saw the jerk argument this page starts from, without the word. On speed, in the 1901 edition of his book:

For the same rolling-stock and for the same comfort in riding, it would seem that a given amount of superelevation must be attained in the same length of time; and hence it is probable that a should vary nearly inversely as the cube of the speed of train.

Arthur N. Talbot, The Railway Transition Spiral, 3rd ed. (1901), §50. His “a” is how fast the curvature grows along the track.

The cube is exactly what constant jerk demands: jerk = v³·dκ/ds, so holding the jerk fixed makes the allowed rate of tightening fall as one over the speed cubed.

Where it is written into the rules today

Britain’s track standard, RSSB’s GCRT5021 (issue six, December 2023), states that “Clothoid spiral transitions are the most commonly used for railway design”, and limits how fast the uncompensated sideways push may change: “The rate of change of cant deficiency shall not exceed the values set out in Table 5”, which for ordinary running is 55 mm/s, exceptionally 70. Cant deficiency is measured as a height across rails taken to be 1500 mm apart, so (our conversion, not the standard’s) 55 mm/s corresponds to a jerk of about 9.81 × 55/1500 ≈ 0.36 m/s³. Where a curve has no transition at all, the standard still has to assume one: it spreads the change over “an assumed distance of 12.2 m”, “the assumed distance between bogie centres”, the length of the vehicle that feels the jump.

Germany’s motorway design guideline (FGSV, RAA 2008) tabulates a minimum clothoid parameter A for each class of road, from 300 m on the fastest (paired with a minimum radius of 900 m) down to 90 m, and requires clothoids with R/3 ≤ A ≤ R on interchange ramps. Since A² = R·L, the fastest class’s minimum means a spiral at least 100 m long into its tightest bend. Not everyone agrees the spiral is worth it on roads: the Nebraska Department of Transportation’s design manual (2022) says it “sees only marginal benefits in the design of spiraled transition curves for new roadway alignments in general”.

Fresnel’s table, and ours

Fresnel computed the bright and dark fringes beside a shadow without a machine, around 1818. His intensities are on a scale where the unobstructed light is 2 (he says to “add to each ½, and finally take the sum of their squares”), so halve them to compare with the instrument above.

Fresnel’s values from the English translation in Henry Crew (ed.), The Wave Theory of Light (1900), p. 124. They agree with a modern 30-digit computation to his fourth decimal place or within a few units of it.

Two things we found wrong along the way

English Wikipedia’s article on the Euler spiral (as read on 23 September 2026) says “By 1880 Arthur Newell Talbot worked out the integral formulas”, citing Levien, who gives 1890; Talbot’s own preface dates his first publication to 1890–91, and 1880 is the year of Holbrook’s article. And Levien’s bibliography lists Talbot’s paper as a reprint “from The Technograph No. 13” of 1899, where Talbot’s preface says it first appeared in No. 5, 1890–91. The 12 g of the Flip Flap, above, is the third.

6. What this page is, and is not

It is a set of exact small models: a car at constant speed on level ground, a pendulum, a frictionless point on a loop, a plane wave past a straight edge. Each shows one mechanism cleanly, and each leaves things out that matter in practice, which the notes under each instrument name: track tilt, damping, vehicle length, friction, and the width and divergence of a real laser beam. The claims that do not depend on the model are the geometric ones. Asking for constant jerk forces the Euler spiral; a circular loop cannot give less than 6 g on entry to a rider sitting in the seat at the top; the edge of a shadow carries exactly a quarter of the light. Nothing here is a design tool for a real road, railway or ride.

The check

Every number above comes from engine.mjs, the file this page runs. The verifier imports that same file (it keeps no copy of its own) and holds it against things computed differently:

The verifier: research/the-bend-that-doesnt-jolt/verify.mjs. Its own test, mutate.mjs, breaks the engine in eight small plausible ways (a flipped sign, a lost factor of two, a loop that forgets gravity) and confirms the verifier fails every time. The 30-digit reference values come from reference.py (mpmath), committed with its output.

Sources

• Raph Levien, The Euler spiral: a mathematical history, Technical Report UCB/EECS-2008-111, University of California, Berkeley (2008). PDF. Source of the Bernoulli, Euler, Fresnel, Cornu, Cesàro, Gravatt, Froude, Holbrook and Higgins material and of the translations quoted; Levien says his Euler quotations are based on Oldfather’s 1933 translation with his own changes.
• Arthur N. Talbot, The Railway Transition Spiral, 3rd ed. (New York: Engineering News Publishing, 1901): definition, §50 on speed, and the preface dating the first publication. Internet Archive.
• Henry Crew (ed.), The Wave Theory of Light: Memoirs by Huygens, Young and Fresnel (1900), p. 124, Fresnel’s table of maxima and minima for exterior fringes. Internet Archive.
• Richard Fitzpatrick, lecture notes on straight-edge diffraction, University of Texas at Austin: the quarter intensity at the shadow edge. Web.
• Ann-Marie Pendrill, “Rollercoaster loop shapes”, Physics Education 40(6), 517–521 (2005), doi:10.1088/0031-9120/40/6/001; quoted from the author’s web version (archived).
• Nick Weisenberger, “Flip Flap Railway”, Coaster101, 28 April 2025: the loop’s 25 ft diameter, the unsourced 12 g, and the 6–8 g estimate. Web.
• IB Stengel GmbH, company timeline. Web. Revolution’s opening date and builder: English Wikipedia, “The Great American Revolution”.
• RSSB, GCRT5021 Track System Requirements, issue six (December 2023), clause 2.6.4.1 and Table 5, guidance G 2.6.1.3–10.
• FGSV, Richtlinien für die Anlage von Autobahnen (RAA 2008), correction sheet of May 2012 (minimum clothoid parameters; ramp rule). PDF.
• Nebraska Department of Transportation, Roadway Design Manual, ch. 3 (May 2022). PDF.
• Reference values: mpmath 1.4.1 at 30 significant digits (reference.py, committed with its output). Fresnel integral conventions: Abramowitz & Stegun, §7.3.

Near here: the force you summon by turning is the sideways push this page ramps; the fastest line through a corner is what a racing driver does with the same v²/r; and the edge of the bow is light piling into a bright edge by geometry, where here it spills past one by waves.