Clothoid

Author:
Will Le
What's the shape of the transitional track connecting a straight track with a circular track such that the transition is smoothest?
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Clothoid is the curve (red) for transition between two tracks (green, blue) of different curvatures (straight line has zero curvature). The two different curvatures are represented by two tangent circles in the figure below (the smaller the circle, the higher curvature).
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When transiting from a lower curvature track to a higher curvature one, in order to avoid shock due to sudden change of centrifugal force, we should increase the curvature gradually (and smoothly). The simplest way is to linearly connect the two curvatures & with the curvature function . Because the centripetal acceleration at a point P on the track is proportional to the curvature at that point ( is the radius of the tangent circle at P), the centrifugal force perceived by passengers will also increases linearly from to (assuming constant velocity ). The original clothoid is the one with linear curvature function, which was studied first by Leonhard Euler (also named "Euler spiral"), then by Marie Alfred Cornu (also named "Cornu spiral"). At last, the name "clothoid" was coined by Ernesto Cesàro following the Greek mythological figure Clotho the spinner who is the first one in the Three Fates (Clotho spins the thread of life, Lachesis draws it out, Atropos cuts it). In general, we can chose any continuous function to express the curvature change (see the drop box "Curvature function" in the sketch above), then we will have a generalized clothoid. There's an interesting family of clothoids generated by the sine curvature functions called "meander curves" resembles the meander of river.
  • When the parameter ϕmax=1, the meander resembles sine curve itself, just a little bit blunter (check "Matching function" to see it).
  • When the parameter ϕmax≈0.766π, the meander closes into a lemniscate (∞).
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When we peel an orange along an equidistant spiral (spherical Archimedean spiral) then flatten it out, we also receive a clothoid, as described in the paper Orange Peels and Fresnel Integrals.
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That double spiral (clothoid) flattened out from spherical equidistant spiral is similar but not the same with the double spiral stereographically projected from spherical equiangular spiral (loxodrome, rhumb line) onto the plane which is a Möbius-transformed logarithmic spiral:
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