Curves

The brachistochrone, tautochrone, Cornu's spiral and the catenary are included


Contents

  1. Introduction
  2. Analytical Theory of Curves
  3. Some Curves
  4. References

Introduction

If you take a pencil and move it with its point constantly in contact with a piece of paper, you'll draw an arbitrary curve on the paper. We might say a "line", but this often means a straight line, a curve drawn to a certain rule that, in fact, has no curvature. Curves drawn according to some rule are more interesting than arbitrary curves, and have attracted the attention of philosophers since early times. In this article, many curves are mentioned, but there are not many pictures, because of the difficulty of making them. The reader is encouraged to draw curves from the hints given, or to consult references with more pictures. We shall restrict ourselves to curves in a plane.

It is more natural to define a curve for our present purposes as the locus of a moving point than as a set of points, since the moving point emphasizes the connectivity and one-dimensional nature of a curve. For some curves, this view is not appropriate. A curve may simply express a mathematical function, such as a continuous probability distribution. The normal distribution is an example of this kind of curve. It is very useful, but not for its properties as a curve. Some properties of a curve as the locus of a moving point are still valuable here for descriptive purposes, however, such as slope, tangent and asymptotes. Even further from our present considerations is the definition of a curve as a set of points. Many pathological curves can be defined as sets, but they have no practical or physical applications. David Hilbert found a continuous curve that passes through all points of an area, for example. For some analytical purposes, however, considering a curve as a continuum of points is valuable. We won't consider these purposes in this article.

Geometry deals overwhelmingly with a particular curve, the straight line. It is defined in Euclid as "that which lies evenly between its extreme points." This is a much better definition than it might seem on the surface. It appeals to the natural straight line of sight, which follows from the propagation of light in a straight line, a "ray". Light is an electromagnetic wave motion, and its rays are normals to the wavefront. In a uniform medium, these rays are straight lines. It is impossible to isolate a single ray, for when the apertures approach the size of the waves of light, diffraction becomes significant. Nevertheless, it is a good approximation for many light phenomena on a human scale. To test whether an edge is straight, a machinist looks along it with his eye, when any deviation becomes obvious. The Euclidean definition of a straight line only idealizes this process. When you look along a line and make its end points coincide, then all other points of the line also coincide. The "end elevation" of a line is a point. The idealization of a line to "length without breadth" is easy.

A simple way to realize a straight line is simply to fold a piece of paper. On opening it out, the fold line will be a straight line. Sometimes we stretch a cord between two supports. At least in a horizontal plane, this defines a straight line between the supports. In a vertical plane, the cord hangs in a catenary. A more common way today to get a straight line is to use a laser. Its beam does spread, but slowly, and represents a straight line to a useful approximation. In geometry, we draw our diagrams with the aid of a straightedge which someone has represented to be one. We can always sight along it to check, or reverse the straightedge and compare.

Another curve that appears prominently in geometry is the circle, the locus of points equidistant from a fixed point, its centre. The constant distance is the radius r. Circles can be drawn with a mechanical device, the compass. Here we do indeed have a curve, one that is uniform in curvature, and we can specify its curvature by the radius, or, better, by the curvature K = 1/r. A zero curvature is an infinite radius, and the curvature increases for smaller and smaller circles. We measure all curvature by comparison with a circle. A straight line may cut the circle in two points (and be, therefore, a secant, or "cutter"), miss the circle entirely, or meet the circle in one point only. In the latter case, the line is a tangent, to which Euclid devotes much attention. He succeeds in finding the properties of a tangent to a circle without limiting processes. More generally, the tangent is the limit of the secant as the two cutting points approach one another. Euclid shows that it is perpendicular to a line from the centre. Such a radial line is a normal to the circle at that point. These concepts of tangent and normal are important in studying any curve.

Analytical Theory of Curves

Although we may specify curves by verbal rules, and study them by the methods of geometry, coordinate or analytical methods are extremely powerful, and have increased our understanding greatly. The curves we know as conic sections were originally defined by rules, then Apollonius showed by geometrical methods that they were also plane sections of a cone, tying them together and revealing many important properties. The same can be done algebraically with analytic geometry, and simply enough that the brilliance of an Apollonius is not required.

Analytical methods were popularized by Descartes, but he did not use the pictorial method with coordinate axes, which arose in the 18th century, and he was not the first to apply algebra to geometry. The reader is, I am sure, quite familiar with rectangular coordinates, which are nevertheless often called Cartesian. The x and y axes are drawn perpendicular, intersecting at the origin O, and graduated from -∞ to +∞. Any point P in the plane is assigned coordinates x and y by drawing lines through P parallel to the axes and noting where they intersect. The origin is the point x = 0, y = 0, or (0,0). If we move our pencil in this plane, each point occupied can be identified by its coordinates (x,y). Our curve may define in this way a function y = f(x) or y = y(x) [we can use y to label the function as well as the variable]. This is called the explicit representation. The plot of a curve on rectangular coordinates is often called a graph. Points can also be specified by polar coordinates (r,θ), where x = r cos θ and y = r sin θ. In this case, the two coordinates are not of the same nature. There are many other ways to specify locations in the plane; these are only very convenient.

Another way to specify a curve analytically is to give x and y as functions of a parameter that varies continuously along the curve. This parameter may be the time t, or distance along the curve s, or anything else. Time and distance are natural parameters. In this case, we have, for example, x = f(t) = x(t), y = g(t) = y(t). We'll use f and g to avoid confusion with x and y, and x and y if we need to make the connection clearer. Even for y = y(x), we may use x as a parameter, whence x = x, y = y(x) are the parametric equations. In these expressions, y(x) is not always the same function, of course, but depends on the parameter chosen. If f(t) and g(t) are continuous for a < t < b, and any point (x,y) is not repeated in this range, then the curve is a simple curve. If x(a) = x(b) and y(a) = y(b) it is a simple closed curve. The term "simple" means that the curve does not intersect itself. The name "Jordan curve" is also used for these curves. A closed curve divides the plane into two parts. As t increases, the area to the right with respect to the sense of motion may be defined the interior of the curve, while that to the left is the exterior. If you decide which is inside and which outside at the beginning, the positive direction of movement along the curve is determined. In many cases, the positive direction of movment along a curve is arbitrary.

A third choice is to specify the curve implicitly, by a function f(x,y) = 0. If we start with y = y(x), then y - y(x) = 0, and our function f(x,y) = y - y(x). Often, it is more convenient to use an implicit representation. For example, a circle of radius a with centre at the origin is conveniently represented by x2 + y2 = a2. A parametric representation is x = a cos t, y = a sin t, and an explicit representation y = ±(a2 - x2)1/2. In this case, we have to use two functions, since f(x) is double-valued. The mode of representation should be chosen to suit the problem posed.

A straight line is represented explicitly by y = mx + b. Clearly, m = Δy/Δx is the slope of the line, the tangent of the angle that the line makes with the x-axis. Calculus tells us that the limit of the Δ-expression is the derivative dy/dx = y', which is useful for any curve. Then, y'(x,y) is the slope of the curve at point (x,y). The tangent line at that point is Y - y = y'(X - x), where (X,Y) is a variable point on the line. The normal line is perpendicular to this, or Y - y = (-1/y')(X - x). The angle θ between two lines of slopes m and m' is given by the formula for the tangent of the difference of two angles as tan θ = (m - m')/(1 + mm'). In this case, 1 + mm' = 1 - 1 = 0, and tan-1 ∞ = 90°, which checks our work.

Suppose ds is a small (infinitesimal) distance along the curve, or arc length. We have assumed implicitly that our curves are rectifiable, or that arc length is well-defined. This is the case if the tangent direction varies smoothly (is piecewise continuous). If ds corresponds to changes dx and dy in the coordinates, then ds = √(dx2 + dy2) = √(1 + y'2) dx, or s - s0 = ∫√(1 + y'2) dx. The lengths of curves can be found by integration in this way. If the curve is specified parametrically, then dx = f'(t)dt and dy = g'(t)dt, so that y' = g'(t)/f'(t). We write f' and g' for the derivatives of these functions with respect to their arguments, so that y' = g'/f' for short, where y' = dy/dx, f' = df/dt and g' = dg/dt.

If the curve is defined by f(x,y) = 0, then if we take the total derivative of f with respect to x we find df/dx = ∂f/∂x + ∂f/∂y (dy/dx) = 0, from which y' = -∂f/∂x/∂f/∂y. The same thing is found if we write df/dy. Now we know how to find dy/dx from each of the ways to express the curve. The derivative is the most important propety of the curve, since it gives the tangent. The tangent is a local approximation to the curve.

Calculus also gives us the curvature. Consider two points on the curve whose abscissae differ by dx. The length of curve between them is ds = √(1 + y'2) dx, and the angle between the tangents at the ends of the segment is Δy'/(1 + y'2), using the formula for the tangent of the difference of two angles and, since the angle is small, equating the difference in tangents to the difference in the angles themselves. If R is the radius of curvature, then RΔθ = Δs, or K = 1/R = y"/(1 + y'2)3/2.

We now need to know how to find y" from the parametric and implicit representations, as we have already found y'. To do this, we differentiate the expressions we have for y' and express the result in terms of the derivatives f',g' of the parametric equations, or the partial derivatives of f(x,y) = 0. The results are y" = (f'g" - g'f")/f'2 and y" = -(fxxfy2 -2fxyfxfy + fyyfx2)/fy3. In the latter expression, fxx means ∂2f/∂x2, and so forth.

For the curvature, we now find K = (f'g" - g'f")/(f'2 + g'2)3/2 or K =(fxxfy2 -2fxyfxfy + fyyfx2)/(fx2 + fy2)3/2. These are complicated expressions, but useful ones. Try them out for the circle r = a.

These expressions for the curvature show that something unusual may occur at points where f' = g' = 0, or fx = fy = 0. When this happens, the expression for K becomes indeterminate. Such points are called singular points. They are not so easily found with the explicit representation y = f(x). A singular point must be further examined to discover what is taking place there. It may be a double point where the curve crosses itself and has two tangents, or a cusp where there is only one tangent and the curve forms a sharp point, or nothing unusual may occur. Points that are not singular are called regular. At regular points, the tangent, as the limit of a secant line, approximates the curve.

For a curve defined explicitly, y = f(x), calculus lets us find maxima and minima, or in general, extrema very easily. If f'(x) = 0, then if f"(x) > 0 we have a maximum and if f"(x) < 0, a maximum. If f"(x) = 0, further investigation is necessary to see whether we have a maximum, minimum or inflection point with a horizontal tangent. If f'(x) ≠ 0, but f"(x) = 0, we have a point of inflection if f" changes sign at x. The curve y = sin x is an example of this, with points of inflection whenever y = 0.

Some Curves

A large class of curves comprises those formed by circles rolling on other circles. A point on a radius of the rolling circle traces the curve. A circle includes a straight line, a circle of infinite radius, and the definition can be extended to the rolling of other shapes than circles, such as ellipses. All of these can in general be called cycloids. The cardioid is the curve generated by a point on the circumference of a circle rolling outside an equal circle. Please refer to the article discussing cardioids on this website for detailed information. The common cycloid is the curve generated by a point on the circumference of a circle rolling on a straight line, usually called simply a "cycloid".

If a circle rolls on the outside of another circle, as for the cardioid, the curve generated is an epicycloid, while if it rolls on the inside, the curve is a hypocycloid. If the point describing the curve is on the radius extended, the curve displays small loops and double points, while if it is on the interior part of the radius, the cusps become smooth depressions. The terms prolate and curtate are sometimes used in these cases. Mechanical devices for drawing cycloids for decorative purposes have been sold as toys. Find the hypocycloid formed by a point on the circumference of a circle of radius a rolling inside a circle of radius 2a. The epicycloid in this case is the nephroid, with two cusps. I think it looks more like a pair of kidneys than one kidney. If you consider a point on a radius of the rolling curve in generating a cardioid that is not on its circumference, the result is a conchoid called the limaçon of Pascal. It is named after Étienne Pascal, the father of Blaise, and its polar equation is r = 2a cos θ + k, where k is a constant.

The cycloid is a good example of the convenience of parametric representation. If θ is the angle through which the circle of radius a has rolled, then it is very easy to conclude that x = a(θ - sin θ) and y = a(1 - cos θ) by looking at a simple diagram. The representations as y = f(x) or f(x,y) = 0 are not nearly as simple, and are rather inconvenient. If the circle rolls with angular velocity ω, then θ = ωt, where t is the time. The derivatives of the parametric equations are f' = a(1 - cos θ) and g' = a sin θ, so that y' = sin θ/(1 - cos θ). When θ = 0, 2π, 4π, ... this expression for y' beomes 0/0, and these points are singular. Using the half-angle formulas, y' = cot θ/2, so the tangent is vertical at these points, and the cycloid has cusps there. At θ = π, 3π, ... , y' = 0, and the tangent is horizontal.

A charged particle moves on a cycloidal path when starting from rest in crossed electric and magnetic fields (see the article on charged particle dynamics under Physics). An inverted cycloid is the brachistochrone, the path of a particle, starting from rest, moving from a point A to a lower point B in minimum time by frictionless sliding. This solution to John Bernoulli's problem enunciated in 1689 surprised his contemporaries. See Courant for details. The period of a cycloidal pendulum is independent of its amplitude, a fact that was used by Huygens in the design of a clock. A cycloidal curve makes an acceptable gear tooth shape, though most gears are now involute. Torricelli wrote a treatise on the cycloid in 1644, and Pascal also studied the curve.

If you start the mass m from any point on the brachistochrone, it will reach the origin O in the same length of time, independently of the starting point. That is, the cycloid is also the tautochrone. If a pendulum swings between cycloidal cheeks, the mass will follow a cycloid, since the involute of a cycloid is another similar cycloid (see below on involutes). The period of the pendulum is the same as that of a small-amplitude simple pendulum of length 4a, where a is the radius of the circle generating the cycloids. If σ is the arc length of the cycloid starting at O, then -dσ/dt is the instantaneous velocity of the mass m. If the mass starts at the point (x,y) on the curve, then conservation of energy gives (m/2)(dσ/dt)2 + mgη = mgy when m is at point (ξ,η). If T' is the time required to reach O from (x,y), then T'√(2g) = ∫(0,y) dσ/√(y - η). If the arc length is given by the function σ = h(η), this can be written as a convolution integral, T'√(2g) = ∫(0,y) (y - η)1/2h'(eta;)dη = y-1/2*h'(y). Taking Laplace transforms, T'√(2g)/s = sH(s)√(π/s), where H(s) is the transform of h(y). Then, sH(s) = T'√(2g)/√(sπ). The inverse transform of this is h'(y) = T'√(2g)/πy1/2. However, h'(y) = dσ/dy = √[1 + (dx/dy)2], so we find the differential equation for the curve, dx/dy = √[(2a - y)/y], where a = gT'22. If we now let y = 2a sin2(θ/2) = a(1 - cos θ)--knowing the answer helps to motivate this substitution--we have dx = a cos2(θ/2) dθ after a little simplification. This integrates easily to x = a(θ + sin θ), so we indeed have a cycloid in parametric representation.

The conventional period of a pendulum, T = 4T', so a is given by a = gT2/16π2. The length of the pendulum L = 4a, so we have L = gT2/4π2, which is the familiar formula for the length of a simple pendulum of period T. The tautochrone gives us Huygens's cycloidal pendulum. The time required to descend the brachistochrone is, then, T/4. The length of a pendulum with T = 2 sec is close to one metre, 99.29 cm in fact. Such a pendulum is said to "beat seconds". It is not difficult to make a cycloidal pendulum by cutting the cheeks with a jigsaw. This will allow you to observe the brachistochrone and tautochrone directly.

The conic sections can be defined by simple rules. An ellipse is the locus of a point the sum of whose distances from two fixed points (the foci) is constant. For an hyperbola, the difference is constant. A parabola is the locus of a point equally distant from a line and a point. These very useful curves are described in articles of their own on this website. They may all be defined by equations f(x,y) = 0 in which the highest power of a variable is 2, and so are quadratic. The general form is ax2 + by2 + cxy + dx + ey + f + g = 0. They have no singular points (but degenerate forms may have double points).

The curve defined as the locus of a point the product of whose distances from two fixed points is constant, equal to the square of half the distance between the points, is the lemniscate of Bernoulli, named by him in 1694. The curve resembles the infinity symbol, and has a singular point at the origin. Implicitly, the curve is defined by (x2 + y2)2 - 2a2(x2 - y2) = 0. The partials are fx = 4x(x2 + y2 - a2) and fy = 4y(x2 + y2 + a2). From these we can find that y' = 0 when x2 + y2 = a2, and that the tangent is vertical at x = ± √2 a. The singular point is x = y = 0. Near this point, the lemniscate is approximately x2 = y2, or x = ± y.

Keeping the same two fixed points, but changing the constant a, we find Cassini's Ovals. For a large value of a, the curve resembles an ellipse. As a decreases, the oval develops bulges at the ends, and for small values of a forms two ovals around the fixed points. Bernoulli's lemsniscate is the boundary case between the bulging oval and the two ovals, when a is half the distance between the points. Like the cardioid, Cassini's ovals and Bernoulli's lemniscate are fourth-order curves.

The semicubic parabola y2 = ax3 has a singular point at the origin. It can be expressed parametrically as x = at2 and y = at3, so f' = 2at and g' = 3at2. Therefore, y' = 3t/2, and the point (0,0), corresponding to t = 0, is singular. This point is a cusp with a horizontal tangent. On the other hand, the cubic parabola y = ax3 has no singular points, since it can be expressed as x = t, y = at3. Since f' = 1, the partials cannot simultaneously vanish. The origin is a regular point with a horizontal tangent where the curvature changes sign, an inflection point.

A related curve is the cissoid, or "ivy-like", from the Greek kissos, "ivy". It is attributed to Diocles. To construct a cissoid, draw a circle of diameter a with its centre on the x-axis, and one end of the diameter at the origin, and draw a vertical line at x = a. Draw any line from the origin intersecting the circle. Then a point M on this line as far from the origin as the point of intersection with the circle is from the vertical line is a point on the cissoid. Its equation is x3 = y2(a - x). For small x, this is an approximate semicubical parabola, but unlike that curve it approaches a vertical asymptote x = a. There is a cusp at the origin. The cissoid can be used to solve the classic problem of the duplication of a cube.

The strophoid, or "turn-like" is yet another third-order curve that is easily constructed geometrically. From the point x = -a draw a line that cuts the y-axis at some point P. Then draw a circle with centre at P and radius OP. The points where this circle intersects the original line are points on the strophoid. At a first glance, this might not seem to work out, but it does, and we find a curve that loops around near the origin and approaches asymptotes. Its equation is y2(a - x) = x2(a + x), and the curve is within -a < x < a. The origin is a singular point. It's easy to see that near the origin (where x is negligible compared to a) the curve is y2 = x2, two straight lines crossing at right angles at the origin. The strophoid, therefore, has a double point at the origin. I can't recall any practical applications of the cissoid or the strophoid.

A spiral is a curve of monotonically changing curvature. Popularly, the term is also used for a helix, which is a quite different thing. The word is from Greek, speira, anything coiled, such as rope. The edges of a French curve are often spirals. The logarithmic spiral is described by the polar equation r = ae, where k is a constant. Polar coordinates are often useful for studying spirals. The usual convention is shown in the figure, where x = r cos θ and y = r sin θ. The angle between the radius to a point and the tangent at the point is given by the expression derived in the figure. For the logarithmic spiral, tan ψ = ae/kae = 1/k, so this angle is constant, and k is the cotangent of the angle ψ. For this reason, it is also called the equiangular spiral. It was investigated by Descartes in 1638.

Another familiar spiral is Archimedes's spiral, in which a point moves with constant speed on a uniformly rotating radial line. The parametric equations are r = vt and θ = ωt. Eliminating t, we find r = (v/ω)θ or r = aθ, where a is a constant, the ratio of the linear to the angular velocities. The hyperbolic spiral has r = a/θ, and r is inversely proportional to θ rather than directly proportional to it. For small θ, this curve begins along an asymptote at a distance a from the origin, then winds in and is asymptotic to the origin.

A string unwinding from a circle describes a spiral known as an involute. The string is always tangent to the circle and normal to the involute, and its length is the radius of curvature. The initial point is a cusp with radial tangent. The locus of the centres of curvature of a curve is called its evolute. Therefore, involutes and evolutes form mutually related pairs. Involute gearing makes use of the property that the normal to the involute, along which the pressure acts, is always tangent to the same circle, and so gives constant velocity. The involute advances radially by 2πa per turn, where a is the radius of the circle. At large distances it is very close to a circle. However, practical uses of the involute use the part of the curve near the generating circle. The involute of a cycloid is another equal cycloid, so the evolute of a cycloid is also a cycloid. This is a remarkable property of the cycloid.

There is a spiral whose curvature is proportional to distance along the curve. A direct attack on the problem of finding this curve is very difficult, but fortunately the solution is known from other approaches, one of which is the integral over a cylindrical wavefront in Fresnel diffraction. This curve is expressed parametrically by x = a√π∫(0,t) cos(πt2/2)dt and y = a√π∫(0,t) sin(πt2/2)dt. The parameter t is the arc length divided by a√π. It is the well-known Cornu's spiral, of which an account can be found in any text on physical optics. The integrals are Fresnel's integrals, and are tabulated, as well as being available from computer programs. The spiral begins at the origin, tangent to the x-axis, and is asymptotic to the point (a√π/2, a√π/2). The other half of the spiral is the same, reflected in the origin. This spiral is also known as the clothoid, from Clotho, the one of the three fates that spins (th pronounced as t). The optical applications do not involve the curvature property.

The Cornu spiral is a vibration curve in optics, showing the contributions to the amplitude from elements of a wavefront. the light intensity is the absolute value squared of the amplitude, while amplitudes are additive. The diagram shows the geometry, with a line source at S and an arbitrary wavefront at radius a. λ is the wavelength of the light. The amplitude at an observation point P is found by summing the contributions from each element ds of the wavefront. The individual contributions are the vibration curve. The Fresnel integrals can be expressed in terms of the dimensionless variable t, defined as in the diagram, and the spiral plotted by x = ∫(0,t)cos(πt2/2)dt and y = ∫(0,t)sin(πt2/2)dt. This spiral has asymptotes at points A=(1/2,1/2) and B=(-1/2,-1/2). The amplitude AB of length √2 corresponds to the undisturbed intensity of the light, which is taken as 2. Other intensities are expressed relative to this. The reader should plot the spiral, or refer to illustrations of it, to make the present discussion clear.

Suppose the observation point P is at the limit of the geometrical shadow of a straight edge. The amplitude there is the vector OA. As we move into the shadow, the amplitude is given by the vector ZA, where Z moves from the origin toward point B, giving a continually decreasing amplitude. As we move into the light, Z moves down the other half of the spiral towards point A, and the amplitude has maxima and minima that form the fringes bounding the shadow. In this way, we can predict the diffraction pattern for apertures and screens bounded by lines parallel to S.

A completely different application of Cornu's spiral is to transition curves in route surveying. The outer rail on a railway curve is raised with respect to the inner rail by e (ft) = gv2/32.2R, where g is the distance between the rails (4.708 ft is standard gauge) and R is the radius of the curve in feet. This superelevation is up to about 8 inches. When the curve is taken at the design speed, the resultant of gravity and centrifugal forces are normal to the track so that the curve is scarcely perceived and the movement is comfortable. To avoid an abrupt change, the superelevation should increase at a constant rate, so the curvature should be proportional to distance as well as the superelevation. Therefore, a transition spiral is introduced between the tangent and the circular curve, perhaps 250 to 500 feet in length. Highways are also superelevated, though here the necessity of a transition spiral is much less, since drivers will automatically compensate.

A mathematically precise curve is not necessary in this application, and convenient approximations are used. With small-angle approximations, Cornu's spiral becomes x = t, y = πt3/6, so a cubic parabola y = ax3 is a good approximation. The curvature in this case is K = 1/R = 6ax. If we want a spiral of length L to join a curve of radius R, then 6aL = 1/R, or a = 1/6LR, so the spiral will be y = x3/6LR. We get a slightly more accurate curve by replacing x by the distance along the curve. Suppose we want a spiral of length L to join a curve of radius R. Then, at the point where the spiral joins the curve, y = L2/6R and y' = L/2R. Making the usual small-angle approximations, we find that if the circular curve is extended backwards until it is parallel to the tangent, this point is halfway along the spiral, at L/2, and the curve is displaced a distance a = L2/24R, as illustrated in the diagram. The spiral connects point TS, where it meets the tangent, and point SC, where it is tangent to the circular curve. This makes it easy to lay out a route with tangents and circular curves, fitting in the spirals later. A transition spiral can always be introduced by reducing the radius of the circular curve slightly. It should be clear that any transition curve without abrupt changes will be adequate in this application.

A catenary is the shape of a uniform, ideally flexible heavy hanging chain. It was named by Huygens in 1691. Applying statics gives us the formula y = a cosh(x/a), with x = 0 at the centre of the chain. The slope is y' = sinh(x/a). The arc length s is very easy to find by integration, since ds/dx = √(1 + y'2) = cosh(x/a). Integration with respect to x gives s = a sinh (x/a). If a cord of length S is stretched between supports a distance L apart, then S = 2a sinh (L/2a) determines the constant a implicitly. The sag of the chain is then s = a[cosh(L/2a) - 1]. If L/2a is small, we see that S = L, approximately. In this case, u = y - a = a[cosh(x/a) - 1] or u = x2/2a, which is a parabola, and s = L2/8a. This gives the curve u = (4s/L2)x2. If the weight is uniformly distributed with x, the parabola is indeed the correct solution for the curve, and not an approximation. In most practical applications, the parabolic curve is the correct one to use.

For a cord supporting a uniform load w per unit x distance, between supports at the same height a distance L apart, the total load is wL, so half of this, wL/2, will be supported at each end. This vertical component V of the tension in the cord decreases linearly from this value to zero at the centre of the span. The horizontal component H is constant. It can be obtained by taking moments about one support for the forces acting on a half-span. This gives Hs = (wL/2)(L/4), or H = wL2/8s, where s is the sag of the span. These results will be approximately true for a catenary where the sag is small compared to the span, and are much easier to handle than the exact results.

For a catenary, the weight wS/2 acts at a distance d which can be found by integrating u cosh u. The result is d = a{(L/S)sinh(L/2a) - (2a/S)[cosh(L/2a) - 1]}, which for small L/2a reduces to L/4. Then, Hs = (wS/2)d or H = wSd/2s, which is constant with respect to x. As a concrete example, consider a chain that is twice as long as the span, or S = 2L. The solution of (L/a) = sinh (L/2a) is L/2a = 2.177319, whence a = L/4.3546. The sag is a[cosh(L/2a) - 1] = 3.4680a = 0.7964L. The vertical force on a half-span acts at a distance d = 0.317L, so the horizontal tension H = 0.199(wS). The total weight of the chain is wS. These results are easily checked experimentally.

The parabola with a span L = 4 and a sag of 3.1856 is y = 0.7964x2, while a catenary of the same dimensions is y = 0.91857[cosh(x/0.91857) - 1]. The length of the catenary is 2L or 8, while the length of the parabola is slightly shorter at 1.9638L or 7.855. The parabola lies above the catenary and is easily distinguished on a plot.

In the practical problem of wire lines, the span L, the weight of the wire per unit length w, and the horizontal tension H are given. We wish to know the sag and the length of wire S per span. For a parabolic curve, we have the result that the sag s = wL2/8H, and from this the other results are easily obtained. If we assume a catenary, then H = (wL/2)[coth(x/2) - 1/x], where x = L/2a. This can be solved with the HP-48G or other method, and then the solution can proceed. To derive the formula, write out H = wS(d/s). The difference between the parabola and the catenary solutions will be very small in practical cases.

Suppose a weight resting on a plane is initially at (0,a) and a cord extends from the weight to the origin. Now let the cord be pulled slowly along the x-axis. The weight will follow a curve called the tractrix. The x-axis is an asymptote, and there is a cusp at (0,a). The equation of this curve is x = a cosh-1(a/y) + √(a2 - y2). Change the + to a - if the weight is dragged to the left. This curve is, surprisingly, the involute of the catenary.

References

R. Courant, Differential and Integral Calculus, Vol. I (London: Blackie and Son, 1936). Chapter V, pp. 258-291.

R. Courant, Differential and Integral Calculus, Vol. II (London: Blackie and Son, 1936). Chapter III, pp. 111-129. The brachistochrone is treated on pp. 491 and 505. It is a more difficult problem than the tautochrone, since it involves an extremum.

Bronshtein and Semendyayev, A Guide Book to Mathematics (Zürich: Harri Deutsch, 1973). pp. 117-131.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 2nd ed. (New York: McGraw-Hill, 1950). Chapter 18. This edition has a particularly clear explanation of how to use Cornu's spiral for Fresnel diffraction at a straight edge. Plots of the spiral are presented, as well as a table of values.

C. F. Allen, Railway Curves and Earthwork, 6th ed. (New York: McGraw-Hill, 1920). Chapter X. The definitive treatment of transition spirals.

The catenary equation sinh(L/2a)=S/2a or sinh x = (S/L)x is easily solved with an HP-48G calculator. Use →SOLVE, choose "solve equation", type in the equation between ' ' delimiters, and press OK. Sinh is found in the MTH HYP menu. Remember to press α before typing x. Put in an initial guess if you want, then press SOLVE to find x = L/2a.


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Composed by J. B. Calvert
Created 18 October 2004
Last revised 7 November 2004