# Euler Spiral

Curve · (x, y, z) = f(u, t)

`(∫₀^(u − π) cos(s² + t) ds,  ∫₀^(u − π) sin(s² + t) ds,  0)`

[Open in the app](https://www.wavelace.com/app#p=110) · [This page](https://www.wavelace.com/presets/euler-spiral)

### What it draws

The first two expressions are integrals, computed numerically at every point: `x = ∫₀^ℓ cos(s² + t) ds` and `y = ∫₀^ℓ sin(s² + t) ds`, with upper limit `ℓ = u − π`. It runs from `−π` to `π` over the one turn of `u`, so both halves are drawn. The third expression is `0`: the curve lies in the vertical `x, y` plane, face on from `Front`.

At `t = 0` these are the Fresnel integrals, `x = C(ℓ)` and `y = S(ℓ)`. The integrand `(cos s², sin s²)` is a unit vector, so `ℓ` is the arc length from the origin and the tangent at `ℓ` points at the angle `ℓ²`. The curvature is therefore `2ℓ`, proportional to the distance travelled: the Euler spiral, or clothoid. It leaves the origin almost straight, bends ever more tightly and coils into a limit point, an *eye*, at `(√(π/8), √(π/8)) ≈ (0.63, 0.63)`. The half with `ℓ < 0` is the same curve turned through `180°`. By `ℓ = π` the tangent has turned through `π² ≈ 9.87` radians, about one and a half times round, and the end sits at `(0.57, 0.77)`, still `0.16` from the eye it circles.

### The clock

Adding `t` inside the cosine and the sine turns every point through the same angle, since `cos(s² + t) = cos s² · cos t − sin s² · sin t`. So `(x, y)` is `(C, S)` rotated by `t`: the spiral spins rigidly about the origin, one radian per second, a full turn every `2π ≈ 6.28` seconds at `Speed` 1.

### History

Leonhard Euler wrote these integrals in 1744, for Jakob Bernoulli's problem of an elastic strip whose curvature grows along its length, and found the eyes in 1781. Alfred Cornu drew the curve in 1874 to read diffraction patterns off a picture, hence its other name, the Cornu spiral. Since the 1890s it has been the transition curve of railways and roads: a bend built on it turns the wheel at a steady rate instead of a jolt.

### Try

- `Rewind` then `Pause`: at `t = 0` the curve is `C(ℓ)` against `S(ℓ)`, the Fresnel integrals themselves.
- Reach further. Make the upper limit `2·(u − π)` in both `x` and `y`: `ℓ` runs to `±2π`, the tangent goes about six times round, and the end closes to `0.08` from its eye.
- Bend faster. Write `cos(2·s^2 + t)` in `x` and `sin(2·s^2 + t)` in `y`: the curvature is `4ℓ`, twice the coils, round eyes at `±(0.44, 0.44)`.
- Lift it. Set `z` to `(u − π)/2`: each point is pushed back by half its arc length, and from `Top` the curve is the graph of `C(ℓ)`, a wave settling to `±0.63`, as in the Fresnel Phasor preset.

### Read more

- [Euler spiral](https://en.wikipedia.org/wiki/Euler_spiral)
- [Fresnel integral](https://en.wikipedia.org/wiki/Fresnel_integral)
- [Curvature](https://en.wikipedia.org/wiki/Curvature)
- [Track transition curve](https://en.wikipedia.org/wiki/Track_transition_curve)