Euler spiral (clothoid)
The curve whose curvature grows in proportion to its length: the transition bend of every motorway and railway.
Adjust parameters
Export
Formulas and key values
- Parametric equations
- Curvature
- The ends wind into
t is the arc length measured from the centre; s is arc length in the curvature formula.
Explainer video
Make a video of the Euler spiral (clothoid)
Opens the Creator with a ready-made brief for this curve. Free-form generation is in beta.
Start the video01 · Definition
What is the Euler spiral (clothoid)?
Leonhard Euler met this curve in 1744 while studying elastic rods; Augustin Fresnel’s integrals describe it, and Marie Alfred Cornu plotted it in 1874 for diffraction, so it is also called the Cornu spiral. Its defining property is simple: the curvature is proportional to the distance travelled along the curve.
That property is why engineers call it the clothoid and use it everywhere. A car driven along it at constant speed turns its steering wheel at a constant rate, so a straight road can blend into a curve without a jolt. Railway and motorway transition curves are pieces of this spiral; at each end it winds into the points (±1/2, ±1/2).
02 · Properties
Key properties
- Curvature proportional to arc length: κ = π·s in this normalisation.
- The parameter t is the arc length measured from the centre.
- Coordinates are the Fresnel integrals C(t) and S(t).
- The two ends wind into the points (1/2, 1/2) and (−1/2, −1/2).
- Used as the transition curve between straight track and circular arcs.
03 · Questions
Euler spiral (clothoid): common questions
- What is the equation of the Euler spiral (clothoid)?
- Parametric equations: x = int₀^t cos (π s^2)/2 ds, y = int₀^t sin (π s^2)/2 ds. t is the arc length measured from the centre; s is arc length in the curvature formula.
04 · Keep exploring