Euler Spiral
(∫₀^(u − π) cos(s² + t) ds, ∫₀^(u − π) sin(s² + t) ds, 0)
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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.