Cornu’s spiral

Cornu’s spiral is the curve parameterized by

x(t) = C(t) = \int_0^t \cos \left( \frac{\pi}{2} s \right) \, ds \\ y(t) = S(t) = \int_0^t \sin \left( \frac{\pi}{2} s \right) \, ds

where C and S are the Fresnel functions, sometimes called the Fresnel cosine integral and Fresnel sine integral. Here’s a plot of the spiral.

Cornu's spiral

Both Fresnel functions approach ½ as t → ∞ and so the curve slowly spirals toward (½, ½) in the first quadrant. And by symmetry, because both functions are odd, the curve spirals toward (-½, -½) in the third quadrant.

Here’s the Python code used to make the plot.

    from scipy.special import fresnel
    from scipy import linspace
    import matplotlib.pyplot as plt

    t = linspace(-7, 7, 1000)
    y, x = fresnel(t)

    plt.plot(x, y)

The SciPy function fresnel returns both Fresnel functions at the same time. It returns them in the order (S, C) so the code reverses the order of these to match the Cornu curve.

One interesting feature of Cornu’s spiral is that its curvature increases linearly with time. This is easy to verify: because of the fundamental theorem of calculus, the Fresnel functions reduce to sines and cosines when you take derivatives, and you can show that the curvature at time t equals πt.

How fast does the curve spiral toward (½, ½)? Since the curvature at time t is πt, that says that at time t the curve is instantaneously bending like a circle of radius 1/πt. So the radius of the spiral is decreasing like 1/πt.

Cornu’s spiral was actually discovered by Euler. Cornu was an engineer who independently discovered the curve much later. Perhaps because Cornu used the curve in applications, his name is more commonly associated with the curve. At least I’ve more often seen it named after Cornu. This is an example of Stigler’s law that things are usually not named after the first person to discover them.

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6 thoughts on “Cornu’s spiral

  1. I’m curious about the engineering application of that.
    If I’m not mistaken, it’s used in road design, because of its constant change in curvature.
    It means that a driver, riding at constant speed, will turn the steering wheel with a constant angular speed. This would avoid the “jerk” on the steering wheel needed to follow a straight-to-circular abrupt transition (that is, a discontinuity in curvature), with the associated “jerk” on car passengers’ bodies due to curvature-dependent centripetal acceleration.
    Is that correct?

  2. In the Feynman Lectures on Physics (chapter 30), he shows how the Cornu Spiral is needed to determine the light intensity at the edge of a shadow. Maybe this was the problem that Cornu was working on when he discovered the curve.

  3. Regarding naming things after Euler, I heard that if all his discoveries were named after him, most of today’s mathematical artifacts would be Euler’s 😉 Not sure where I read about it though…

  4. I would add the explicit comment that the spiral converges to {-1/2,-1/2} when t goes to -infinity. I know the “symmetry” comment was probably meant to hint to that but it wasn’t very clear if the symmetry applies to t or to xy.

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