Orange Peels and Fresnel Integrals

There are two standard ways of peeling an orange: either cut the skin along meridians, or cut it along a spiral. We consider here the second method, and study the shape of the spiral strip, when unfolded on a table. We derive a formula that describes the corresponding flattened-out spiral. Cutting the peel with progressively thinner strip widths, we obtain a sequence of increasingly long spirals. We show that, after rescaling, these spirals tends to a definite shape, known as the Euler spiral. The Euler spiral has applications in many fields of science. In optics, the illumination intensity at a point behind a slit is computed from the distance between two points on the Euler spiral. The Euler spiral also provides optimal curvature for train tracks between a straight run and an upcoming bend. It is striking that it can be also obtained with an orange and a kitchen knife.

There are two standard ways of peeling an orange: either cut the skin along meridians, or cut it along a spiral. We consider here the second method, and study the shape of the spiral strip, when unfolded on a table. We derive a formula that describes the corresponding flattened-out spiral. Cutting the peel with progressively thinner strip widths, we obtain a sequence of increasingly long spirals. We show that, after rescaling, these spirals tends to a definite shape, known as the Euler spiral. The Euler spiral has applications in many fields of science. In optics, the illumination intensity at a point behind a slit is computed from the distance between two points on the Euler spiral. The Euler spiral also provides optimal curvature for train tracks between a straight run and an upcoming bend. It is striking that it can be also obtained with an orange and a kitchen knife. A natural breakfast question, for a mathematician, is which shape the spiral peel will have, when flattened out. We derive a formula that, for a given cut width, describes the corresponding spiral's shape.
For the analysis, we parametrize the spiral curve by a constant speed trajectory, and express the curvature of the flattened-out spiral as a function of time. This is achieved by comparing a revolution of the spiral on the orange with a corresponding spiral on a cone tangent to the surface of the orange ( fig. 3, left). Once we know the curvature, we derive a differential equation for our spiral, which we solve analytically ( fig. 4, left).
We then consider what happens to our spirals when we vary the strip width. Two properties are affected: the overall size, and the shape. Taking finer and finer widths of strip, we obtain a sequence of increasingly  long spirals; rescale these spirals to make them all of the same size. We show that, after rescaling, the shape of these spirals tends to a well defined limit. The limit shape is a classical mathematical curve, known as the Euler spiral or the Cornu spiral ( fig. 4, right). This spiral is the solution of the Fresnel integrals.
The Euler spiral has many applications. In optics, it occurs in the study of light diffracting through a slit [1, §10.3.8]. More precisely, the illumination intensity at a point behind a slit is the square of the distance between two points on the Euler spiral, easily determined from the slit's geometry.
. left: Spiral on the sphere, transferred to the tangent cone, and developed on the plane, for computing its radius of curvature; right: The computation of the radius of curvature R of the flattened spiral.
The same spiral is also used in civil engineering: it provides optimal curvature for train tracks [4, §14.1.2]. A train that travels at constant speed and increases the curvature of its trajectory at a constant rate will naturally follow an arc of the Euler spiral. The review [2] describes the history of the Euler spiral and its three independent discoveries.

ANALYSIS
For the purpose of our mathematical treatment, we shall replace the orange by a sphere of radius one. The spiral on the sphere is taken of width 1/N , see ( fig. 1). The area of the sphere is 4π, so the spiral has a length of roughly 4πN . We describe the flattenedout orange peel spiral by a curve (x(t), y(t)) in the plane, parameterized at unit-speed from time t = −2πN to t = 2πN .
On a sphere of radius one, the area between two horizontal planes at heights h 1 and h 2 is 2π(h 2 − h 1 ), see ( fig. 5). It follows that, at time t, the point on the sphere has height s := t/2πN . Our first goal is to find a differential equation for (x(t), y(t)). For that, we compute the radius of curvature R(t) of the flattened-out spiral at time t: this is the radius of circle with best contact to the curve at time t. For example, R(−2πN ) = R(2πN ) = 0 at the poles, and R(0) = ∞ at the equator.
For N large, the spiral at time t follows roughly a parallel at height s on the orange. The surface of the sphere can be approximated by a tangent cone whose development on the plane is a disk sector ( fig. 3,  left). The radius of that disk equals the radius of curvature of the spiral at time t, and can be computed using Thales' theorem ( fig. 3, right). The radius R(t) is in fact only determined up to sign; our choice reflects the NE-SW orientation of the spiral on the sphere. Now, the condition that we move move at unit speed on the sphere -and on the plane -is (ẋ) 2 + (ẏ) 2 = 1, and the condition that the spiral has a curvature of R(t) isẋÿ −ẍẏ = 1/R. Here,ẋ andẏ are the speeds of x and y respectively, andẍ andÿ are their accelerations. In fact, introducing the complex path z(t) = x(t) + iy(t), the conditions can be expressed as |ż| 2 = 1 andzż = i/R.

The solution has the general form
for a real function φ; indeed, its derivative is computed asż = exp(iφ(t)) and has norm 1. Aszż = iφ(t), we haveφ(t) = s/ √ 1 − s 2 , which has as elementary solution φ(t) = − (2πN ) 2 − t 2 . We have deduced that the flattened-out spiral has parameterization The flattened-out peel of an orange is shown in (fig. 2), and the corresponding analytic solution, computed by MAPLE [3], is shown in (fig. 4, left). The orange's radius was 3cm, and the peel was 1cm wide, giving N = 3.

LIMITING BEHAVIOUR
What happens if N tends to infinity, that is, if we peel the orange with an ever thinner spiral? For that, we recall the power series approximation , which we substitute with a = 2πN in the above expression: Taking only values of N that are integers, this simplifies to The approximation error is defined by the condition that the radius of curvature at time t is 1/2t; here the parameterization is over t from −∞ to +∞. The corresponding curve is called the Euler spiral and winds infinitely often around the points ±( π 8 , π 8 ). Setting T := t/ √ 4πN , the condition |t| N 0.7 becomes |T | N 0.2 . We have thus proven: Theorem. If T N 0.2 , then the part of the orange peel of width 1/N parameterized between − √ 4πN T and √ 4πN T is a good approximation for the part of the Euler spiral parameterized between −T and T .
Note that for large N , the piece of the orange peel parameterized between − √ 4πN T and √ 4πN T forms a rather thin band around the orange's equator. The contribution of rest of the orange disappears due to the rescaling process.

CONCLUSION
The Euler spiral is a well known mathematical curve. In this article, we explained how to construct it with an orange and a kitchen knife. Flattened fruit peels have already been considered, e.g. those of apples [5], but were never studied analytically. The Euler spiral that we obtained has had many discoveries across history [2]; ours occurred over breakfast.