Plane Curves

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AML710 CAD LECTURE 10

PLANE CURVES

1. Analytical curves 2. Synthetic Curves

PLANE CURVES The curves are an important part of many engineering disciplines. We recognise curves entirely confined to a plane as plane curves as opposed to the curves existing in 3D spaces. The curves arise because of the fact that we don’t have only polyhedral objects in the real world but there are many curved objects also. To describe such curved objects and their boundaries we need different types of curves. A curve may result as a solution of an algebraic equation f(x,y)=0 in a plane or as the solution of an equation like g(x,y,z)=0 in space.

1 ANALYTIC CURVES Mathematically the curves can be described using algebraic equations or in terms of a parameter (parametric). The analytical curves can be ‹ Explicit y= f(x) => for every x there is one value Eg: y=mx+C Can we represent a circle (or any closed curve) by such equation? ‹ Implicit (multi valued or closed functions) f(x,y) = 0 => multi-valued or closed curves Eg: x2 + y 2 − r 2 = 0 x2 y 2 + =1 a2 b2

Curve Fitting and Curve Fairing When we have a given set of points through which a curve has to pass through, essentially we are fitting a curve to the data points. An analytic equation can be written for the curve we have fit through this data. Any point on the curve can be determined by interpolation

On the other hand, the curve fairing technique is used when an approximate relation of curve is sufficient instead of an accurate one. We use least square or some other method to minimize the y^2 deviations. Such techniques give an approximate curve b bx 2 n y = ax ; y = ae ; y = c1 + c2 x + c3 x +...+ cn+1x

2 Curve Fitting Here a curve fitting is done using polynomials of order n-3 to n-1 where n is the number of points

The order of the polynomial used directly influences the shape of the curve

ANALYTICAL AND PARAMETRIC REPRESENTATIONS Analytical Representations ‹ Precision ‹ Compact Storage – we store only the equations ‹ Ease of Interpolation ‹ Slope and radius of curvature can be determined easily ‹ Any point on the curve can be precisely determined

Parametric Representations ‹ The slope of the curve is represented by tangent vectors ‹ Infinite slope results when one of the components of tangent vector is zero ‹ As parameter is used, parametric representation is independent of axis ‹ The curve end points and length are fixed by the range

3 PARAMETRIZATION Parametrization is a way to write a function so that all the coordinates (or variables) depend on the same variable. Example: If we have a function z = f[x,y], and if the parameter is "t" where the x-coordinate is expressible as g[t], and the y-coordinate is expressible as h[t], we say we can write the function coordinate-wise as {x[t], y[t], z[t]}. We reduce the problem of two variables to that of one input variable (t).

Consider the parametrization of a unit circle x^2+y^2=1 as: P(t) =[x y] where x = x(t)& y = y(t) Is there unique way of parametrizing the given function?

PARAMETRIC CURVES In parametric form each coordinate of a point is represented as a function of a single parameter, say t. P(t) =[x y] where x = x(t)& y = y(t) The derivative or tangent vector on the curve is given by P'(t) = [x'(t) y'(t)] The slope of the curve dy/dx – dy dy / dt y'(t) = = dx dx / dt x'(t) Infinite slope results when one of the components of tangent vector is zero As a single parameter is used, parametric representation is independent of axis The curve end points and length are fixed by the parametric range

4 Example of parametric curves A simplest parametric curve, a straight line in single parameter t, is given as

P(t)= P1 + (P2 − P1)t 0 ≤ t ≤1 The components of P(t) in parametric form are:

x(t) = x1 + (x2 − x1)t 0 ≤ t ≤1

y(t) = y1 + (y2 − y1)t Ex: Determine the line segment between the position vectors (1 2) (4 3). Also determine the slope and tangent vector.

P(t) = P1 + (P2 − P1)t = (1 2)+ [(4 3)− (1 2)]t = (1 2)+ (3 1)t

x(t) = x1 + (x2 − x1)t =1+ 3t;

y(t) = y1 + (y2 − y1)t = 2 + t The tangent vector and slope are: P'(t) = [x'(t) y'(t)]= [3 1] = 3i + j dy dy / dt y'(t) = = =1/ 3 dx dx / dt x'(t)

CONIC SECTIONS The general second-degree equation for conic sections is ax2 + 2bxy + cy 2 + 2dx + 2ey + f = 0

5 CONIC SECTIONS: Projective geometry The general second-degree equation for conic sections is ax2 + 2bxy + cy 2 + 2dx + 2ey + f = 0

Taken from Geometry by M. Audin

CONIC SECTIONS The general second-degree equation for conic sections is ax2 + 2bxy + cy 2 + 2dx + 2ey + f = 0 By defining coefficients a, b, c, d, e and f we get variety of conic sections. If the curve is defined with respect to a local coordinates and passes through origin, then f=0. Suitable geometric boundary conditions are used to establish curves through specific points Eg.: c=1, we need 5 independent B.Cs to define the curve between two points. These could be 1. Position of two end points, 2. Slope at these points and 3. Another intermediate point through which the curve must pass.

6 CONIC SECTIONS Ellipse/circle and hyperbola are central conics and parabola is not. Ellipse and circle form a compact whereas parabola and hyperbola extend to infinity. How can we draw an ellipse? Hint: An ellipse with foci F and F’ is locus of a point M such that (MF + MF’)=2a for some non-negative number a such that 2a > FF’

Source: mathworld.wolfram.com

Representation of a circle A non-parametric representation of a circular arc in the first quadrant is

y = ± 1− x2 0 ≤ x ≤1 ‹ Parametric representation

π P(θ ) = [x y]= [cosθ sinθ ] 0 ≤ θ ≤ 2 ‹ Alternatively the following parametric form can be used 1− t 2 2t x = ; y = 0 ≤ t ≤1 1+ t 2 1+ t 2

7 Algorithmic Representation of a circle What is the problem with the scheme just discussed? Computationally intensive because of repeated calculations What is the solution? ‹ An improved algorithm due to L.B.Smith (1969)

xi+1 = r cos(θi +δθ ), yi+1 = r sin(θi +δθ )

Where θi is the parameter that determines (xi yi)

xi+1 = r(cosθi cosδθ − sinθi sinδθ );

yi+1 = r(cosθi sinδθ + cosδθ sinθi ) We know that

xi = r cosθi ; yi = r sinθi This results in

xi+1 = xi cosδθ − yi sinδθ; 2π yi+1 = xi sinδθ + yi cosδθ whereδθ = (n−1)

Representation of a circle Notes: 1. The quantityδθ is constant. Therefore cosδθ and sinδθ need to be calculated only once and stored. 2. Only 6 inner loop operations are involved (a) 4 multiplications (b) one subtraction (c) one addition 3. The result is comparable to the simple parametric representation while achieving better efficiency 4. A non-origin centered circle can be generated by translating the origin centered circle. ‹ Alternatively a unit circle can be used to obtain any circle of desired radius by suitably scaling and translation.

8 Parametric Representation of an ellipse Simple relations: x = a cosθ; y = bsinθ 0 ≤ θ ≤ 2π (1) ‹ An improved algorithm due to I.B.Smith (1969)

xi+1 = a cos(θi +δθ ), yi+1 = bsin(θi +δθ )

Where θi is the parameter that determines (xi yi)

xi+1 = a(cosθi cosδθ − sinθi sinδθ );

yi+1 = b(cosθi sinδθ + cosδθ sinθi )

Substitute from (1) xi = a cosθi ; yi = bsinθi

This results in a xi+1 = xi cosδθ − b yi sinδθ ); b 2π yi+1 = a xi sinδθ + yi cosδθ ) whereδθ = (n−1)

Representation of an ellipse Notes: 1. If a=b it reduces to the case of a circle 2. quantity δθ is constant. Therefore need to be calculated only once. 2. Only 8 inner loop operations are involved (a) 6 multiplications (b) one subtraction (c) one addition 3. The result is comparable to the simple parametric representation while achieving better efficiency 4. A non-origin centered ellipse can be generated by translating the origin centered ellipse.

9 Properties of a parametric Ellipse & circle • The arc length s, curvature k, and tangential angle phi of the circle with radius r represented parametrically are • Arc length: s(t) = rt (1) • Curvature: k(t) =1/ t (2) • Tangential angle: φ(t) = t / r (3) b2 • Eccentricity e = 1− 0 ≤ e ≤1 (4) a2 • Perimeter (approx.) p = π 2(a2 + b2 ) (5)

Properties of a parametric curves • For any parametric curve: P(t) = [x(t) y(t)] Speed of parameterization: v(t) = P′(t) = x′(t) y′(t) Eg: Consider a circle of radius r P(t) = [r cos(t) r sin(t)] x′ = −r sin(t) y′ = r cos(t) v(t) = x′(t)2 + y′(t)2 = r cos2 (t) + sin 2 (t) = r A parameterization is called regular iff v(t) ≠ 0

10 Arc length of parametric curves • For any parametric curve: P(t) = [x(t) y(t)] With Speed of parameterization: v(t) = P′(t) = x′(t) y′(t) The arc length of the curve is defined as: t t L(t) = — v(t)dt = — x′2 (t) + y′(t)dt 0 0 For the case of circle the arc length is: t 2π 1 L(t) = — v(t)dt = —[r 2 sin 2 (t) + r 2 cos2 (t)]2 dt = 2πr 0 0

‹ Representation of an ellipse %Example 4.4 (R&A) %Ellipse a=4, b=1 inclined 30 deg. to horizontal center at (2 2) n=33; 1.5 dt=2*pi/32; a=4; 1 b=1; 0.5 sdt=sin(dt); cdt=cos(dt); 0 xo=1.0; yo=1.0; -0.5 for i=1:33 -1 x(i)=xo*cdt-a/b*yo*sdt; y(i)=xo*b/a*sdt+yo*cdt; -1.5 -5 -4 -3 -2 -1 0 1 2 3 4 5 xo=x(i); yo=y(i); end plot(x,y)

11 ‹ Some Exercises ‹ 1. The circumcircle of an ellipse, i.e., the circle whose center concurs with that of the ellipse and whose radius is equal to the ellipse's semimajor axis.

‹ 2. An ellipse intersects a circle in 0, 1, 2, 3, or 4 points. The points of intersection of a circle of center (x_0,y_0) and radius r with an ellipse of semi-major and semi-minor axes a and b, respectively and center (x_e,y_e) can be determined by simultaneously solving ‹ (x-x_0)^2+(y-y_0)^2=r^2 (1) ‹ ((x-x_e)^2)/(a^2)+((y-y_e)^2)/(b^2)=1. (2)

‹ Some Exercises

‹ If (x_0,y_0)=(x_e,y_e)=(0,0), then the solution takes on the particularly simple form ‹ x = +/-a*sqrt((r^2-b^2)/(a^2-b^2)) (3) ‹ y = +/-b*sqrt((a^2-r^2)/(a^2-b^2)). (4)

12 ‹ Parametric Representation of a Parabola Analytical form: y2 = 4ax Parametric form: 2 π x = tan φ; y = ±2 a tanφ 0 ≤ φ ≤ 2 (1) ‹ An improved algorithm due to L.B.Smith (1969)

2 x = aθ , y = 2aθ 0 ≤ θ ≤ ∞ 18 16

14 Where θ is the parameter that determines (x y) 12 10 Since it is an open curve we need to calculate the 8 6 limits to display the curve 4

0 0 20 40 60 80 100 120 140 160 180 200 xmin xmax θmin = a ; θmax = a based on x

ymin ymax θmin = 2a ; θmax = 2a based on y

‹ Parametric Representation of a Parabola

Starting with x = aθ 2 , y = 2aθ 0 ≤ θ ≤ ∞ (1)

2 2 xi+1 = aθi + 2aθiδθ − a(δθ ) ;

yi+1 = 2aθi + 2aθiδθ 18 16 Substituting from (1) 14 12 2 10 xi+1 = xi + yiδθ + a(δθ ) ; 8 6 2π y = y + 2x δθ whereδθ = 4 i+1 i i (n−1) 2

0 0 20 40 60 80 100 120 140 160 180 200

13 ‹ Parametric Representation of a Hyperbola Analytical form: x2 y 2 − =1 a2 b2 π Parametric form: x = ±asecθ, y = ±b tanθ 0 ≤ θ ≤ 2 (1) Recall the identities sec(θ + δθ ) = 1/ cos(θ + δθ ) =1/(cosθ cosδθ − sinθ sinδθ ) tan(θ + δθ ) = (tanθ + tanδθ ) /(1− tanθ tanδθ ) Expanding sum of angles ab / cosθ x = ±asec(θ + δθ ) = ± ; i+1 bcosδθ − b tanθ sinδθ b tanθ + b tanδθ y = ±a tan(θ +δθ ) = ± i+1 1− tanθ tanδθ

Substituting from (1) bxi xi+1 = ± ; bcosδθ − yi sinδθ

b(yi + b tanδθ ) 2π yi+1 = ± whereδθ = (n−1) b − yi tanδθ

‹ Parametric Representation of a Hyperbola Alternate Method Recall the hyperbolic functions x = a coshθ,= (eθ + e−θ ) y = bsinhθ = (eθ − e−θ ) 0 ≤ θ ≤ ∞ (1) 150 Expanding sum of angles

100 xi+1 = a(coshθi coshδθ − sinhθi sinhδθ ); y = b(coshθ sinhδθ + coshδθ sinhθ ) i+1 i i 50 Substituting from (1)

a 0 xi+1 = xi coshδθ − b yi sinhδθ; 0 10 20 30 40 50 60 b 2π yi+1 = a xi sinhδθ + yi coshδθ whereδθ = (n−1) The limits are −1 θmin = cosh (xmin / a); −1 −1 2 θmax = cosh (xmax / a) where cosh (x) = ln(x + x −1)

14 ‹ Parametric Representation of other plane curves ‹ Confocal conics

‹ Elliptic gears

Source: mathworld.wolfram.com

‹ Parametric Representation of other plane curves ‹ Epicycloid

a + b x = (a + b)cosφ − bcos( φ) b a + b y = (a + b)sinφ − bsin( φ) b

Source: mathworld.wolfram.com

15 ‹ Parametric Representation of other plane curves ‹ Hypocycloid

a − b x = (a − b)cosφ − bcos( φ) b a − b y = (a − b)sinφ − bsin( φ) b

Source: mathworld.wolfram.com

‹ Parametric Representation of other plane curves ‹ Gear Curve

Here a=1, b=10 and n=1-12 x = r cost y = r sin t 1 r = a + tanh[bsin(nt)] b Source: mathworld.wolfram.com

16 Chapter 2 Famous Plane Curves

Plane curves have been a subject of much interest beginning with the Greeks. Both physical and geometric problems frequently lead to curves other than ellipses, parabolas and hyperbolas. The literature on plane curves is extensive. Diocles studied the cissoid in connection with the classic problem of doubling the cube. Newton1 gave a classification of cubic curves (see [Newton] or [BrKn]). Mathematicians from Fermat to Cayley often had curves named after them. In this chapter, we illustrate a number of historically-interesting plane curves. Cycloids are discussed in Section 2.1, lemniscates of Bernoulli in Section 2.2 and cardioids in Section 2.3. Then in Section 2.4 we derive the differential equation for a catenary, a curve that at first sight resembles the parabola. We shall present a geometrical account of the cissoid of Diocles in Section 2.5, and an analysis of the tractrix in Section 2.6. Section 2.7 is devoted to an illustration of clothoids, though the significance of these curves will become

Sir Isaac Newton (1642–1727).English mathematician, physicist, and astronomer. Newton’s contributions to mathematics encompass not only his fundamental work on calculus and his discovery of the binomial theorem for negative and fractional exponents, but also substantial work in algebra, number theory, classical and analytic geometry, methods of com- putation and approximation, and probability. His classiﬁcation of cubic curves was published as an appendix to his book on optics; his work in analytic geometry included the introduction of polar coordinates. 1 As Lucasian professor at Cambridge, Newton was required to lecture once a week, but his lectures were so abstruse that he frequently had no auditors. Twice elected as Member of Parliament for the University, Newton was appointed warden of the mint; his knighthood was awarded primarily for his services in that role. In Philosophiæ Naturalis Principia Mathematica, Newton set forth fundamental mathematical principles and then applied them to the development of a world system. This is the basis of the Newtonian physics that determined how the universe was perceived until the twentieth century work of Einstein.

39 40 CHAPTER 2. FAMOUS PLANE CURVES clearer in Chapter 5. Finally, pursuit curves are discussed in Section 2.8. There are many books on plane curves. Four excellent classical books are those of Ces`aro [Ces], Gomes Teixeira2 [Gomes], Loria3 [Loria1], and Wieleitner [Wiel2]. In addition, Struik’s book [Stru2] contains much useful information, both theoretical and historical. Modern books on curves include [Arg], [BrGi], [Law], [Lock], [Sav], [Shikin], [vonSeg], [Yates] and [Zwi].

2.1 Cycloids

The general cycloid is deﬁned by

cycloid[a,b](t)= at b sin t, a b cos t . − − Taking a = b gives cycloid[a,a], which describes the locus of points traced by a point on a circle of radius a which rolls without slipping along a straight line.

Figure 2.1: The cycloid t (t sin t, 1 cos t) 7→ − −

In Figure 2.2, we plot the graph of the curvature κ2(t) of cycloid[1, 1](t) over the range 0 6 t 6 2π. Note that the horizontal axis represents the variable t, and not the x-coordinate t sin t of Figure 2.1. A more faithful representation − of the curvature function is obtained by expressing the parameter t as a function 1 of x. In Figure 2.3, the curvature of a given point (x(t),y(t)) of cycloid[1, 2 ] is represented by the point (x(t), κ2(t)) on the graph vertically above or below it.

2 Francisco Gomes Teixeira (1851–1933). A leading Portuguese mathematician of the last half of the 19th century. There is a statue of him in Porto.

Gino Loria (1862–1954). Italian mathematician, professor at the Univer- 3 sity of Genoa. In addition to his books on curve theory, [Loria1] and [Loria2], Loria wrote several books on the history of mathematics. 2.1. CYCLOIDS 41

-1 1 2 3 4 5 6 7

-1

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-7

Figure 2.2: Curvature of cycloid[1, 1] 4

2 4 6 -1

-2

1 Figure 2.3: cycloid[1, 2 ] together with its curvature

Consider now the case in which a and b are not necessarily equal. The curve cycloid[a,b] is that traced by a point on a circle of radius b when a concentric circle of radius a rolls without slipping along a straight line. The cycloid is prolate if ab.

Figure 2.4: The prolate cycloid t (t 3 sin t, 1 3cos t) 7→ − − 42 CHAPTER 2. FAMOUS PLANE CURVES

Figure 2.5: The curtate cycloid t (2t sin t, 2 cos t) 7→ − −

The ﬁnal ﬁgure in this section demonstrates the fact that the tangent and normal to the cycloid always intersect the vertical diameter of the generating circle on the circle itself. A discussion of this and other properties of cycloids can be found in [Lem, Chapter 4] and [Wagon, Chapter 2].

Figure 2.6: Properties of tangent and normal

Instructions for animating Figures 2.4–2.6 are given in Notebook 2. 2.2. LEMNISCATES OF BERNOULLI 43

2.2 Lemniscates of Bernoulli

Each curve in the family a cos t a sin t cos t (2.1) lemniscate[a](t)= , , 1 + sin2 t 1 + sin2 t 4 is called a lemniscate of Bernoulli . Like an ellipse, a lemniscate has foci F1 and F2, but the lemniscate is the locus L of points P for which the product of 2 distances from F1 and F2 is a certain constant f . More precisely,

2 L = (x, y) distance (x, y), F1 distance (x, y), F2 = f , | where distance(F1, F2)=2f. This choice ensures that the midpoint of the segment connecting F1 with F2 lies on the curve L . Let us derive (2.1) from the focal property. Let the foci be ( f, 0) and let ± L be a set of points containing (0, 0) such that the product of the distances from F1 = ( f, 0) and F2 = (f, 0) is the same for all P L . Write P = (x, y). − ∈ Then the condition that P lie on L is

(2.2) (x f)2 + y2 (x + f)2 + y2 = f 4, − or equivalently (2.3) (x2 + y2)2 =2f 2(x2 y2). − The substitutions y = x sin t, f = a/√2 transform (2.3) into (2.1). Figure 2.7 displays four lemniscates. Starting from the largest, each succes- sively smaller one passes through the foci of the previous one. Figure 2.8 plots the curvature of one of them as a function of the x-coordinate.

Figure 2.7: A family of lemniscates

Jakob Bernoulli (1654–1705). Jakob and his brother Johann were the ﬁrst of a Swiss mathematical dynasty. The work of the Bernoullis was instru- 4 mental in establishing the dominance of Leibniz’s methods of exposition. Jakob Bernoulli laid basic foundations for the development of the calculus of variations, as well as working in probability, astronomy and mathematical physics. In 1694 Bernoulli studied the lemniscate named after him. 44 CHAPTER 2. FAMOUS PLANE CURVES

Although a lemniscate of Bernoulli resembles the ﬁgure eight curve parametrized in the simper way by

(2.4) eight[a](t)= sin t, sin t cos t , a comparison of the graphs of their respective curvatures shows the diﬀerence between the two curves: the curvature of a lemniscate has only one maximum and one minimum in the range 0 6 t < 2π, whereas (2.4) has three maxima, three minima, and two inﬂection points.

-1 -0.5 0.5 1

-1

-2

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Figure 2.8: Part of a lemniscate and its curvature

In Section 6.1, we shall deﬁne the total signed curvature by integrating κ2 through a full turn. It is zero for both (2.1) and (2.4); for the former, this follows because the curvature graphed in Figure 2.8 is an odd function.

Figure 2.9: A family of cardioids 2.3. CARDIOIDS 45

2.3 Cardioids

A cardioid is the locus of points traced out by a point on a circle of radius a which rolls without slipping on another circle of the same radius a. Its parametric equation is

cardioid[a](t)= 2a(1+ cos t)cos t, 2a(1 + cos t) sin t .

The curvature of the cardioid can be simpliﬁed by hand to get 3 κ2[cardioid[a]](t)= . 8 a cos(t/2) | | The result is illustrated below using the same method as in Figures 2.3 and 2.8. The curvature consists of two branches which meet at one of two points where the value of κ2 coincides with the cardioid’s y-coordinate.

Figure 2.10: A cardioid and its curvature

2.4 The Catenary

In 1691, Jakob Bernoulli gave a solution to the problem of finding the curve assumed by a flexible inextensible cord hung freely from two fixed points; Leibniz has called such a curve a catenary (which stems from the Latin word catena, meaning chain). The solution is based on the differential equation

2 d2y 1 dy (2.5) = 1+ . dx2 a dx s 46 CHAPTER 2. FAMOUS PLANE CURVES

To derive (2.5), we consider a portion pq of the cable between the lowest point p and an arbitrary point q. Three forces act on the cable: the weight of the portion pq, as well as the tensions T and U at p and q. If w is the linear density and s is the length of pq, then the weight of the portion pq is ws.

Θ q

p T

Figure 2.11: Deﬁnition of a catenary

Let T and U denote the magnitudes of the forces T and U, and write | | | | U = U cos θ, U sin θ , | | | | with θ the angle shown in Figure 2.11. Because of equilibrium we have

(2.6) T = U cos θ and ws = U sin θ. | | | | | | Let q = (x, y), where x and y are functions of s. From (2.6) we obtain dy ws (2.7) = tan θ = . dx T | | Since the length of pq is x 2 dy s = 1+ dx, dx s Z0 the fundamental theorem of calculus tells us that

2 ds dy (2.8) = 1+ . dx dx s When we diﬀerentiate (2.7) and use (2.8), we get (2.5) with a = ω/ T . | | 2.5. CISSOID OF DIOCLES 47

Although at ﬁrst glance the catenary looks like a parabola, it is in fact the graph of the hyperbolic cosine. A solution of (2.5) is given by x (2.9) y = a cosh . a The next ﬁgure compares a catenary and a parabola having the same curvature at x = 0. The reader may need to refer to Notebook 2 to see which is which. 8

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Figure 2.12: Catenary and parabola

We rotate the graph of (2.9) to deﬁne t (2.10) catenary[a](t)= a cosh , t , a where without loss of generality, we assume that a > 0. A catenary is one of the few curves for which it is easy to compute the arc length function. Indeed,

0 t catenary[a] (t) = cosh , | | a and it follows that a unit-speed parametrization of a catenary is given by

s2 s s a 1+ 2 , arcsinh . 7→ r a a! In Chapter 15, we shall use the catenary to construct an important minimal surface called the catenoid.

2.5 The Cissoid of Diocles

The cissoid of Diocles is the curve deﬁned nonparametrically by (2.11) x3 + xy2 2ay2 =0. − 48 CHAPTER 2. FAMOUS PLANE CURVES

To find a parametrization of the cissoid, we substitute y = xt in (2.11) and obtain 2at2 2at3 x = , y = . 1+ t2 1+ t2 Thus we define 2at2 2at3 (2.12) cissoid[a](t)= , . 1+ t2 1+ t2 The Greeks used the cissoid of Diocles to try to find solutions to the problems of doubling a cube and trisecting an angle. For more historical information and the definitions used by the Greeks and Newton see [BrKn, pages 9–12], [Gomes, volume 1, pages 1–25] and [Lock, pages 130–133]. Cissoid means ‘ivy-shaped’. Observe that cissoid[a]0(0) = 0 so that cissoid is not regular at 0. In fact, a cissoid has a cusp at 0, as can be seen in Figure 2.14. The definition of the cissoid used by the Greeks and by Newton can best be explained by considering a generalization of the cissoid. Let ξ and η be two curves and A a fixed point. Draw a line ` through A cutting ξ and η at points Q and R respectively, and find a point P on ` such that the distance from A to P equals the distance from Q to R. The locus of such points P is called the cissoid of ξ and η with respect to A.

Ξ Η

AQPR

Figure 2.13: The cissoid of ξ and η with respect to the point A

Then cissoid[a] is precisely the cissoid of a circle of radius a and one of its tangent lines with respect to the point diametrically opposite to the tangent line, as in Figure 2.14. Let us derive (2.11). Consider a circle of radius a centered at (a, 0). Let (x, y) be the coordinates of a point P on the cissoid. Then 2a = distance(A, S), so by the Pythagorean theorem we have 2.5. CISSOID OF DIOCLES 49

A S

Figure 2.14: AR moves so that distance(A, P )= distance(Q, R)

distance(Q,S)2 = distance(A, S)2 distance(A, Q)2 (2.13) − 2 = 4a2 distance(A, R) distance(Q, R) . − − The deﬁnition of the cissoid says that

distance(Q, R)= distance(A, P )= x2 + y2.

By similar triangles, distance(A, R)/(2a)= distancep(A, P )/x. Therefore, (2.13) becomes 2 2a (2.14) distance(Q,S)2 =4a2 1 (x2 + y2). − x − On the other hand,

distance(Q,S)2 = distance(R,S)2 distance(Q, R)2 − = distance(R,S)2 distance(A, P )2 (2.15) − 2 2ay = (x2 + y2), x − since by similar triangles, distance(R,S)/(2a)= y/x. Then (2.11) follows easily by equating the right-hand sides of (2.14) and (2.15). 50 CHAPTER 2. FAMOUS PLANE CURVES

From Notebook 2, the curvature of the cissoid is given by 3 κ2[cissoid[a]](t)= , a t (4 + t2)3/2 | | and is everywhere strictly positive.

-1 -0.5 0.5 1

Figure 2.15: Curvature of the cissoid

2.6 The Tractrix

A tractrix is a curve α passing through the point A = (a, 0) on the horizontal axis with the property that the length of the segment of the tangent line from any point on the curve to the vertical axis is constant, as shown in Figure 2.16.

Figure 2.16: Tangent segments have equal lengths 2.6. TRACTRIX 51

The German word for tractrix is the more descriptive Hundekurve. It is the path that an obstinate dog takes when his master walks along a north-south path. One way to parametrize the curve is by means of t (2.16) tractrix[a](t)= a sin t, cos t + log tan 2 , It approaches the vertical axis asymptotically as t 0 or t π, and has a cusp → → at t = π/2. To find the differential equation of a tractrix, write tractrix[a](t) = (x(t),y(t)). Then dy/dx is the slope of the curve, and the differential equation is therefore dy √a2 x2 (2.17) = − . dx − x It can be checked (with the help of (2.19) overleaf) that the differential equation is satisfied by the components x(t)= a sin t and y(t)= a cos t + log(tan(t/2)) of (2.16), and that both sides of (2.17) equal cot t.

Hx,yL x

Ha,0L

Figure 2.17: Finding the diﬀerential equation

The curvature of the tractrix is

κ2[tractrix[1]](t)= tan t . −| | In particular, it is everywhere negative and approaches at the cusp. Figure −∞ 2.18 graphs κ2 as a function of the x-coordinate sin t, so that the curvature at a given point on the tractrix is found by referring to the point on the curvature graph vertically below. 52 CHAPTER 2. FAMOUS PLANE CURVES

For future use let us record Lemma 2.1. A unit-speed parametrization of the tractrix is given by s ae−s/a, 1 e−2t/a dt for 0 6 s< , − ∞ 0 (2.18) α(s)=  Zs p  s/a  ae , 1 e2t/a dt for

0 1 (2.19) tractrix[a] (φ)= a cos φ, sin φ + . − sin φ Deﬁne φ(s)= π arcsin(e−s/a) for s > 0. Then sin φ(s)= e−s/a; furthermore, − π/2 6 φ(s) < π for s > 0, so that cos φ(s)= 1 e−2s/a. Hence − − −s/a 0 e psin φ(s) (2.20) φ (s)= = . a√1 e−2s/a −a cos φ(s) −

-1

-2

~~Figure 2.18: A tractrix and its curvature 2.7. CLOTHOIDS 53~~

~~Therefore, if we deﬁne a curve β by β(s)= tractrix[a] φ(s) , it follows from (2.19) and (2.20) that 0 0 0 β (s) = tractrix[a] φ(s) φ (s)~~

1 sin φ(s) = a cos φ(s), sin φ(s)+ − sin φ(s) −a cos φ(s) = sin φ(s), cos φ(s) = e−s/a, 1 e−2s/a = α0(s). − − − − Also, β(0) = (a, 0) = α(0). Thus α and β coincidep for 0 6 s < , so that ∞ α is a reparametrization of a tractrix in that range. The proof that α is a reparametrization of tractrix[a] for < s 6 0 is similar. Finally, an easy −∞ calculation shows that α has unit speed.

~~2.7 Clothoids~~

One of the most elegant of all plane curves is the clothoid or spiral of Cornu5. We give a generalization of the clothoid by deﬁning t un+1 t un+1 clothoid[n,a](t)= a sin du, cos du . n +1 n +1 Z0 Z0 Clothoids are important curves used in freeway and railroad construction (see [Higg] and [Roth]). For example, a clothoid is needed to make the gradual transition from a highway, which has zero curvature, to the midpoint of a freeway exit, which has nonzero curvature. A clothoid is clearly preferable to a path consisting of straight lines and circles, for which the curvature is discontinuous. The standard clothoid is clothoid[1,a], represented by the larger one of two plotted in Figure 2.19. The quantities t t sin(πu2/2)du and cos(πu2/2)du, Z0 Z0 are called Fresnel6 integrals; clothoid[1,a] is expressible in terms of them. Since ±∞ ±∞ √π sin(u2/2)du = cos(u2/2)du = ± 2 Z0 Z0

5Marie Alfred Cornu (1841–1902). French scientist, who studied the clothoid in connection with diﬀraction. The clothoid was also known to Euler and Jakob Bernoulli. See [Gomes, volume 2, page 102–107] and [Law, page 190].

~~6 Augustin Jean Fresnel (1788–1827). French physicist, one of the founders of the wave theory of light. 54 CHAPTER 2. FAMOUS PLANE CURVES~~

(as is easily checked by computer), the ends of the clothoid[1,a] curl around the points 1 a√π(1, 1). The ﬁrst clothoid is symmetric with respect to the origin, ± 2 but the second one (smaller in Figure 2.19) is symmetric with respect to the horizontal axis. The odd clothoids have shapes similar to clothoid[1,a], while the even clothoids have shapes similar to clothoid[2,a].

~~1.5~~

~~0.5~~

~~-1.5 -1 -0.5 0.5 1 1.5~~

~~-0.5~~

-1

~~-1.5~~

~~1 Figure 2.19: clothoid[1, 1] and clothoid[2, 2 ]~~

Although the definition of clothoid[n,a] is quite complicated, its curvature is simple: tn (2.21) κ2[clothoid[n,a]](t)= . − a In Chapter 5, we shall show how to define the clothoid as a numerical solution to a differential equation arising from (2.21).

~~2.8 Pursuit Curves~~

The problem of pursuit probably originated with Leonardo da Vinci. It is to ﬁnd the curve by which a vessel moves while pursuing another vessel, supposing that the speeds of the two vessels are always in the same ratio. Let us formulate this problem mathematically.

Deﬁnition 2.2. Let α and β be plane curves parametrized on an interval [a,b]. We say that α is a pursuit curve of β provided that (i) the velocity vector α0(t) points towards the point β(t) for a

~~0 (ii) the speeds of α and β are related by α0 = k β , where k is a positive k k k k constant. We call k the speed ratio.~~

~~A capture point is a point p for which p = α(t1)= β(t1) for some t1.~~

~~In Figure 2.20, α is the curve of the pursuer and β the curve of the pursued.~~

~~Figure 2.20: A pursuit curve~~

When the speed ratio k is larger than 1, the pursuer travels faster than the pursued. Although this would usually be the case in a physical situation, it is not a necessary assumption for the mathematical analysis of the problem. We derive diﬀerential equations for pursuit curves in terms of coordinates.

Lemma 2.3. Write α = (x, y) and β = (f,g), and assume that α is a pursuit curve of β. Then (2.22) x02 + y02 = k2(f 02 + g02) and (2.23) x0(y g) y0(x f)=0. − − − 0 Proof. Equation (2.22) is the same as α0 = k β . To prove (2.23), we k k k k observe that α(t) β(t) = x(t) f(t),y(t) g(t) and α0(t) = x0(t),y0(t) . − − − Note that the vector J α(t) β(t) = y(t)+g(t), x(t) f(t) is perpendicular − − − to α(t) β(t). The condition that α0(t) is a multiple of α(t) β(t) is conveniently − − expressed by saying that α0(t) is perpendicular to J α(t) β(t) , which is − equivalent to (2.23). Next, we specialize to the case when the curve of the pursued is a straight line. Assume that the curve β of the pursued is a vertical straight line passing through the point (a, 0), and that the speed ratio k is larger than 1. We want to ﬁnd the curve α of the pursuer, assuming the initial conditions α(0) = (0, 0) and α0(0) = (1, 0). 56 CHAPTER 2. FAMOUS PLANE CURVES

~~We can parametrize β as~~

~~β(t)= a, g(t) .~~

~~Furthermore, the curve α of the pursuer can be parametrized as~~

~~α(t)= t, y(t) .~~

~~The condition (2.22) becomes~~

~~(2.24) 1+ y02 = k2g02, and (2.23) reduces to (y g) y0(t a)=0. − − − Diﬀerentiation with respect to t yields~~

~~(2.25) y00(t a)= g0. − − From (2.24) and (2.25) we get~~

0 00 (2.26) 1+ y 2 = k2(a t)2y 2. − Let p = y0; then (2.26) can be rewritten as k dp dt = . 1+ p2 a t − This separable ﬁrst-order equationp has the solution 1 a t (2.27) arcsinh p = log − , −k a when we make use of the initial condition y0(0) = 0. Then (2.27) can be rewritten as y0 = p = sinh arcsinh p = 1 earcsinh p e−arcsinhp 2 − −1 1 1 a t /k a t /k = − − . 2 a − a ! Integrating, with the initial condition y(0) = 0, yields

1+ 1 1− 1 ak 1 ak a t /k ak a t /k (2.28) y = + − − . k2 1 2 k +1 a − k 1 a − − ! The curve of the pursuer is then α(t) = (t, y(t)), where y is given by (2.28). Since α(t1)= β(t1) if and only if t1 = a, the capture point is ak (2.29) p = a, . k2 1 − 2.9. EXERCISES 57

~~The graph below depicts the case when a = 1 and k has the values 2, 3, 4, 5. As the speed ratio k becomes smaller and smaller, the capture point goes higher and higher.~~

~~k=5~~

~~yHtL~~

~~k=4~~

~~0.6~~

~~k=3 0.5~~

~~k=2 0.4~~

~~0.3~~

~~0.2~~

~~0.1~~

~~t 0.2 0.4 0.6 0.8 1~~

~~Figure 2.21: The case in which the pursued moves in a straight line~~

~~2.9 Exercises~~

M 1. Graph the curvatures of the cycloids illustrated in Figures 2.4 and 2.5. Find the formula for the curvature κ2 of the general cycloid cycloid[a,b]. Then deﬁne and draw ordinary, prolate and curtate cycloids together with the deﬁning circle such as those on page 42.

~~M 2. A deltoid is deﬁned by~~

deltoid[a](t)= 2a cos t(1 + cos t) a, 2a sin t(1 cos t) . − − The curve is so named because it resembles a Greek capital delta. It is a particular case of a curve called hypocycloid (see Exercise 13 of Chapter 6). Plot as one graph the deltoids deltoid[a] for a = 1, 2, 3, 4. Graph the curvature of the ﬁrst deltoid. 58 CHAPTER 2. FAMOUS PLANE CURVES

~~M 3. The Lissajous7 or Bowditch curve8 is deﬁned by~~

~~lissajous[n,d,a,b](t)= a sin(nt + d), b sin t .~~

~~Draw several of these curves and plot their curvatures. (One is shown in Figure 11.19 on page 349.)~~

M 4. The limac¸on, sometimes called Pascal’s snail, named after Etienne´ Pascal, father of Blaise Pascal9, is a generalization of the cardioid. It is deﬁned by limacon[a,b](t)=(2a cos t + b) cos t, sin t .

~~Find the formula for the curvature of the lima¸con, and plot several of them.~~

5. Consider a circle with center C = (0,a) and radius a. Let ` be the line tangent to the circle at (0, 2a). A line from the origin O = (0, 0) intersecting ` at a point A intersects the circle at a point Q. Let x be the ﬁrst coordinate of A and y the second coordinate of Q, and put P = (x, y). As A varies along ` the point P traces out a curve called versiera, in Italian and mis- named in English as the witch of Agnesi10. Verify that a parametrization of the Agnesi versiera is

~~agnesi[a](t)= 2a tan t, 2a cos2 t . Jules Antoine Lissajous (1822–1880). French physicist, who studied sim- 7 ilar curves in 1857 in connection with his optical method for studying vibrations.~~

Nathaniel Bowditch curve (1773–1838). American mathematician and astronomer. His New American Practical Navigator, written in 1802, was highly successful. Bowditch also translated and annotated Laplace’s 8 Mécanique Céleste. His study of pendulums in 1815 included the figures named after him. Preferring his post as president of the Essex Fire and Marine Insurance Company from 1804 to 1823, Bowditch refused chairs of mathematics at several universities.

Blaise Pascal (1623–1662). French mathematician, philosopher and inven- 9 tor. Pascal was an early investigator in projective geometry and invented the ﬁrst mechanical device for performing addition and subtraction.

Maria Gaetana Agnesi (1718–1799). Professor at the University of 10 Bologna. She was the ﬁrst woman to occupy a chair of mathematics. Her widely used calculus book Instituzioni Analitiche was translated into French and English. 2.9. EXERCISES 59

M 6. Deﬁne the curve cos t sin t tschirnhausen[n,a](t)= a , a . (cos(t/3))n (cos(t/3))n When n = 1, this curve is attributed to Tschirnhausen11. Find the formula for the curvature of tschirnhausen[n,a][t] and make a simultaneous plot of the curves for 1 6 n 6 8.

~~7. In the special case that the speed ratio is 1, show that the equation for the pursuit curve is~~

~~2 a a t a t y(t)= − 1 2log − , 4 a − − a ! and that the pursuer never catches the pursued.~~

~~M 8. Equation (2.28) deﬁnes the function~~

~~ak k(a t)1+ 1/k a1/kk(a t)1− 1/k y(t)= + − − . k2 1 2a1/k(1 + k) − 2(k 1) − − Plot a pursuit curve with a = 1 and k =1.2.~~

11Ehrenfried Walter Tschirnhausen (1651–1708). German mathematician, who tried to solve equations of any degree by removing all terms except the ﬁrst and last. He contributed to the rediscovery of the process for making hard-paste porcelain. Sometimes the name is written von Tschirnhaus.