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Orange Peels and Fresnel Integrals

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Orange Peels and Fresnel Integrals Orange Peels and Fresnel Integrals LAURENT BARTHOLDI AND ANDRE´ HENRIQUES ut the skin of an orange along a thin spiral of The same spiral is also used in civil engineering: it constant width (Fig. 1) and place it flat on a table provides optimal curvature for train tracks between a CC(Fig. 2). A natural breakfast question, for a mathe- straight run and an upcoming bend [4, §14.1.2]. A train that matician, is what shape the spiral peel will have when travels at constant speed and increases the curvature of its flattened out. We derive a formula that, for a given cut trajectory at a constant rate will naturally follow an arc of width, describes the corresponding spiral’s shape. the Euler spiral. For the analysis, we parametrize the spiral curve by a The review [2] describes the history of the Euler spiral constant speed trajectory, and express the curvature of the and its three independent discoveries. flattened-out spiral as a function of time. For the purpose of our mathematical treatment, we shall This is achieved by comparing a revolution of the spiral replace the orange by a sphere of radius one. The spiral on on the orange with a corresponding spiral on a cone tan- the sphere is taken to be of width 1/N, as in Fig. 5. The area gent to the surface of the orange (Fig. 3, left). Once we of the sphere is 4p, so the spiral has a length of roughly 4p know the curvature, we derive a differential equation for N. We describe the flattened-out orange-peel spiral by a our spiral, which we solve analytically (Fig. 4, left). curve (x(t), y(t)) in the plane, parametrized at unit speed We then consider what happens to our spirals when we from time t =-2pN to t = 2pN. vary the strip width. Two properties are affected: the On a sphere of radius 1, the area between two hori- overall size, and the shape. Taking finer and finer widths of zontal planes at heights h1 and h2 is 2p(h2 - h1) (see strip, we obtain a sequence of increasingly long spirals; Fig. 5). It follows that, at time t, the point on the sphere has rescale these spirals to make them all of the same size. We height s :¼ t=2pN : show that, after rescaling, the shape of these spirals tends Our first goal is to find a differential equation for to a well-defined limit. The limit shape is a classical (x(t), y(t)). For that, we compute the radius of curvature mathematical curve, known as the Euler spiral or the Cornu R(t) of the flattened-out spiral at time t: this is the radius of spiral (Fig. 4, right). This spiral is the solution of the Fresnel the circle with best contact to the curve at time t. For integrals. example, R(-2pN) = R(2pN) = 0 at the poles, and Rð0Þ¼ The Euler spiral has many applications. In optics, it 1 at the equator. occurs in the study of light diffracting through a slit [1, For N large, the spiral at time t follows roughly a parallel §10.3.8]. Let light shine through a long and thin horizontal at height s on the orange. The surface of the sphere can be slit and hit a vertical wall just behind it; assume that the slit’s approximated by a tangent cone whose development on width and the distance to the wall are comparable to the the plane is a disk sector (Fig. 3, left). The radius qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wavelength. What is the illumination intensity on the wall, pffiffiffiffiffiffiffiffiffiffiffiffi as a function of height? Imagine two ants running after each RðtÞ¼ 1 À s2=s ¼ ð2pN Þ2 À t2=t other on the Euler spiral at constant speed. It turns out that the square of the distance between the ants, at time t,is of that disk equals the radius of curvature of the spiral at proportional to the illumination intensity on the wall at time t (Fig. 3, right). The radius R(t) is in fact only deter- height t. mined up to sign; the construction on one hemisphere Partially supported by the Courant Research Centre ‘‘Higher Order Structures’’ of the University of Go¨ ttingen. Ó 2012 The Author(s). This article is published with open access at Springerlink.com, Volume 34, Number 3, 2012 1 DOI 10.1007/s00283-012-9304-1 Figure 3. (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 Figure 1. An orange, assumed to be a sphere of radius 1, and of the flattened spiral. spiral of width 1/N, with N = 3. 6 1 4 0.5 2 0 0 -2 -0.5 -4 -6 -1 -6 -4 -2 0 2 4 6 -1 -0.5 0 0.5 1 Figure 4. (left) Maple plot of the orange peel spiral (N = 3). (right) The Euler spiral: lim N ? ? produces a spiral curling one way, and curling the opposite way on the other. Now, the condition that we move at unit speed on the sphere — and on the plane — is ðx_Þ2 þðy_Þ2 ¼ 1, and the Figure 2. The orange peel in Figure 1 now flattened out. condition that the spiral has a curvature of R(t)is ......................................................................................................................................................... LAURENT BARTHOLDI is Professor of ANDRE´ HENRIQUES is Professor at the Mathematics at the University of Go¨ttingen. 2 University of Utrecht. He is a topologist at Apart from recreational mathematics, he is heart, even though his research has been AUTHORS interested in group theory, abstract algebra, scattered across various areas of mathematics, and complex dynamics. He enjoys playing including combinatorics, representation the- the piano and harpsichord, and, when in ory, category theory, and operator algebras. mountainous areas, the alphorn. He recently spent a year away from his research in order to teach math classes at a Mathematiches Institut local high school. Georg-August Universita¨t zu Go¨ttingen Go¨ttingen Mathematisch Instituut Germany Universiteit Utrecht e-mail: [email protected] Utrecht The Netherlands e-mail: [email protected] 2 THE MATHEMATICAL INTELLIGENCER The above curve is, up to scaling and parametrization speed, the solution of the classical Fresnel integral Z Z t t ðXðtÞ; Y ðtÞÞ ¼ cos u2du; sin u2du ; 0 0 defined by the condition that the radius of curvature at time t is 1/2t; here the parametrization is over t from 1 to þ1. The corresponding curve is called the Eulerpffiffi p spiralffiffi and p p winds infinitelypffiffiffiffiffiffiffiffiffi often around the points ð 8; 8Þ. Setting T :¼ t= 4pN, the condition |t| N 0.7 becomes |T| N 0.2. We have thus proven 0.2 THEOREM If T N , then the partpffiffiffiffiffiffiffiffiffi of the orangepffiffiffiffiffiffiffiffiffi peel of Figure 5. Area of a thin circular strip on the sphere. width 1/N parametrized between À 4pNT and 4pN Tis a good approximation for the part of the Euler spiral x_y€ À x€y_ ¼ 1=R. Here, x_ and y_ are the derivatives of x and y, parametrized between -T and T (which corresponds to respectively, and € and € are their second derivatives. In 2 x y T /2p revolutions on each side of the spiral). fact, introducing the complex path z(t) = x(t) + iy(t), the conditions can be expressed as jz_j2 ¼ 1 and z€z_ ¼ i=R. Note that for large Nffiffiffiffiffiffiffiffiffi, the piece offfiffiffiffiffiffiffiffiffi the orange peel param- The solution has the general form p p Z etrized between À 4pN T and 4pN T forms a rather thin t band (of width N -0.3) around the orange’s equator. The zðtÞ¼ expði/ðuÞÞdu; aforementioned approximations we made do not apply to 0 the rest of the orange. When the part of the peel contained for a real function /; indeed, its derivative is computed as in that thin band is unrolled, it covers up to N0.4 revolutions z_ ¼ expði/ðtÞÞ and has norm 1. As z€z_ ¼ i/_ ðtÞ, we have, pffiffiffiffiffiffiffiffiffiffiffiffi of the Euler spiral. _ 2 substituting the expression for 1/R, /ðtqÞ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis= 1 À s , As a consequence, the Euler spiral is the limit shape of a wide class of flattened peels: if you take any (convex) fruit which has as elementary solution /ðtÞ¼ ð2pN Þ2 À t2. that looks like a sphere in a neighborhood of its equator, We have deduced that the flattened-out spiral has param- then the flattened peel of that fruit will tend to the Euler etrization spiral as the peel becomes thinner. More generally, we 8 Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t conjecture that any smooth convex body with positive > 2 < xðtÞ¼ cos ð2pN Þ À u2du; curvature will share that same property. Z0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t The Euler spiral is a well-known mathematical curve. In > 2 : yðtÞ¼ sin ð2pN Þ À u2du: this article, we explained how to construct it with an orange 0 and a kitchen knife. Flattened fruit peels have already been The flattened-out peel of an orange is shown in Fig. 2, and considered, for example, those of apples [5], but they were the corresponding analytic solution, computed by MAPLE [3], never studied analytically. The Euler spiral has had many is shown in Fig. 4, left. The orange’s radius was 3 cm, and discoveries across history [2]; ours occurred over breakfast. the peel was 1 cm wide, yielding N = 3. What happens if N tends to infinity, that is, if we peel the OPEN ACCESS orange with an ever thinner spiral? For that, we recall the This article is distributed under the terms of the Creative power series approximation Commons Attribution License which permits any use, dis- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 u4 a2 À u2 ¼ a À þO ; tribution, and reproduction in any medium, provided the 2a a3 original author(s) and the source are credited.
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