CONIC SECTIONS DAVID PIERCE

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1 CONIC SECTIONS DAVID PIERCE Contents List of Figures 1 1. Introduction 2 2. Background Definitions Motivation 3 3. Equations Focus and directrix The poar equation Lines through the focus Distances Areas The rectanguar equations 12 Bibiography 14 References 14 List of Figures 1 A conic surface and cone 2 2 The method of finding a mean proportiona 3 3 Two mean proportionas 4 4 Construction of points of the paraboa 4 5 Conic sections of different eccentricities 5 6 Derivation of the poar equation of a conic section 7 7 The eipse 8 8 The paraboa 8 9 The hyperboa 9 10 Extreme points in the eipse and paraboa 9 11 Extreme points in the hyperboa Conic sections as determined by equations of areas Conjugate hyperboas The circe as a imit of conics 16 Date: January 3,

2 2 DAVID PIERCE 1. Introduction These notes are about the pane curves known as conic sections. The mathematica presentation is mainy in the anaytic stye whose origins are sometimes said to be the Geometry [7] of René Descartes. However, the features of conic sections presented in 3 beow were apparenty known to mathematicians of the eastern Mediterranean in ancient times. Accordingy, 2 beow contains a review what I have been abe to find out about the ancient knowedge. I try to give references to the origina texts or transations of them. Meanwhie, I ist some reevant approximate dates; the ancient dates are seected from [5, pp. 685 f.]: 350: Menaechmus on conic sections; 300: Eucid, Eements; 225: Apoonius, Conics; 212: death of Archimedes; +320: Pappus, Mathematica Coections; +560: Eutocius, commentaries on Archimedes; +1637: Descartes, Geometry. The reader of these notes may agree that the conic sections are worthy of study, independenty of any appication. However, Isaac Newton , for exampe, coud not have deveoped his theory of gravitation [8] without knowing what the Ancients knew about conic sections Background 2.1. Definitions. A cone and its associated conic surface are determined by the foowing data: 1 a circe, caed the base of the cone; 2 a point, caed the vertex of the cone and the conic surface; the vertex must not ie in the pane of the base. The conic surface consists of the points on the ines that pass through the vertex and the circumference of the base. The cone itsef is the soid figure bounded by the surface and the base. See Figure 1. vertex base Figure 1. A conic surface and cone 1 An inverse-square aw of gravitation causes panetary orbits to be conic sections. Newton showed this, apparenty using such knowedge as can be found in Apoonius. It may be that Newton inferred, from ancient secondary sources, that the ancient scientists themseves were aware of an inverse-square aw of gravity [9, 11.7].

3 CONIC SECTIONS 3 The definitions of cone and conic surface can be found at the beginning of the treatise On Conic Sections [1, 2, 3, 11], by Apoonius of Perga. 2 The axis of the cone is the ine joining the vertex to the center of the base. There is no assumption that the axis is perpendicuar to the base; if it is, then the cone is right; otherwise, the cone is obique. A conic section is the intersection of a pane with a conic surface. The discovery of conic sections as objects worthy of study is generay 3 attributed to Apoonius s predecessor Menaechmus. However, there are three kinds of conic sections: the eipse, the paraboa, and the hyperboa. According to Eutocius [11, pp ], Apoonius was the first mathematician to show that each kind of conic section can be obtained from every conic surface. Indeed, the names of the three kinds of conic sections appear [11, p. 283 f., n. a] to be due to Apoonius as we. The names are meaningfu in Greek and refect the different geometric properties of the sections, in a way shown in Motivation. Menaechmus used conic sections to sove the probem of dupicating the cube. Suppose a cube is given, with voume V ; how can a cube be constructed with voume 2V? We can give a symboic answer: If the side of the origina cube has ength s, then the new cube must have side of ength s 3 2. But how can a side of that ength be constructed? The corresponding probem for squares can be soved as foows. Suppose AB is a diameter of a circe, and C is on AB, and D is on the circumference of the circe, and CD AB. Then the square on CD is equa in area to the rectange whose sides are AC and BC. More symboicay, if engths are as in Figure 2, then a x = x b, or ab = x2, 1 so that x 2 a = b 2 a. 2 In particuar, if b/a = 2, then a square with side of ength x has area twice that of a square with side of ength a. D x A a C b B Figure 2. The method of finding a mean proportiona Suppose instead we have Then a x = x y = y b. 3 x 3 a 3 = x a y x b y = b a. 4 2 Perga or Perge was near what is now Antaya; its remains are we worth a visit. 3 See for exampe [11, p. 283 f., n. a] or [6, p. 1].

4 4 DAVID PIERCE If b/a = 2, then a cube with side of ength x has voume twice that of a cube with side of ength a. In any case, the severa engths can be arranged as in Figure 3. There, ange ACB is right, and BCD and ACE are diameters of the indicated circes. D F A a x C y E b B Figure 3. Two mean proportionas The probem is, How can D and E be chosen on the extensions of BC and AC so that the circes intersect as in Figure 3? The soution of Menaechmus aong with many other soutions is given in the commentary [4, pp ] by Eutocius on the second voume On the Sphere and the Cyinder by Archimedes. In Figure 3, if CDF E is a rectange, then F determines x and y. But by Equations 3, rearranged, x and y must satisfy two equations, ay = x 2, bx = y 2. 5 Each of these equations determines a curve, and F is the intersection of the two curves. The curves turn out to be conic sections, namey paraboas. Points on the curve given by ay = x 2 can be potted as in Figure 4. Figure 4. Construction of points of the paraboa If one imagines that the circes in Figure 4 are not a in the same pane, but serve as parae bases of cones bounded by the same conic surface, then one may be abe to see how the curve arises as a section of that surface. However, an aternative approach to the conic sections was given by Pappus of Aexandria [10, p ]; it may have been

5 CONIC SECTIONS 5 due originay to Eucid of Aexandria, athough his works on conic sections are ost. We can take the aternative approach as foows. 3. Equations 3.1. Focus and directrix. A conic section ζ is determined by the foowing data: 1 a ine d, caed the directrix of ζ; 2 a point F not on d, caed the focus of ζ; 3 a positive rea number or distance e, caed the eccentricity of ζ. Then ζ comprises the points P in the pane of d and F such that P F = e P d. 6 Some exampes are in Figure 5, with the same directrix and focus, but various eccentricities. The exampes are drawn by computer by means of 25 beow. See aso Figure Figure 5. Conic sections of different eccentricities Suppose we assign a rectanguar coordinate system to the pane of ζ in which F has the coordinates h, k, and d is defined by Ax + By + C = 0 7

6 6 DAVID PIERCE where A 0 or B 0. Then ζ is defined by Ax + By + C x h2 + y k 2 = e, 8 A2 + B 2 hence aso by x h 2 + y k 2 = e 2 Ax + By + C2. 9 A 2 + B 2 This equation is not very usefu for showing the shape of ζ. By choosing the rectanguar coordinate system appropriatey, we can ensure h, k = 0, 0, B = 0, A = 1, C > Then C is the distance between the focus and the directrix, and 9 becomes x 2 + y 2 = e 2 x + C The poar equation. Equation 11 is nicer than 9, but is sti not the most usefu rectanguar equation for ζ. However, 11 becomes more usefu when converted to poar form. Reca the conversion-equations: { x = r cos θ, r 2 = x 2 + y 2, y = r sin θ; tan θ = y x. 12 So the poar form of 11 is which is equivaent to r 2 = e 2 r cos θ + C 2, 13 ±r = er cos θ + C. 14 The pus-or-minus sign here is needed, uness we know that r aways has the sign of r cos θ + C, or aways has the opposite sign. It does not. However, note we that the same point can have different poar coordinates; in particuar, the same point has poar coordinates r, θ and r, θ + π. We sha use this fact frequenty. The equation r = er cos θ + C 15 is equivaent to Hence, if s, ϕ satisfies 15, then s, ϕ + π satisfies r = e r cosθ + π + C. 16 r = er cos θ + C. 17 So we can take either 15 or 17 as the poar equation for ζ. We can aso derive 14 directy from the origina definition of ζ; see Figure 6. We can rewrite 17 as r = er cos θ + ec, 18 r er cos θ = ec, 19 r1 e cos θ = ec. 20 Since ec 0, the factor 1 e cos θ wi never be 0, so we can divide by it, obtaining r = ec 1 e cos θ. 21

7 CONIC SECTIONS 7 C + r cos θ P d P P F r d C O F θ r cos θ Q r = ec + r cos θ Figure 6. Derivation of the poar equation of a conic section If we rewrite 15 the same way, we get ec r = 1 e cos θ. 22 Again, either 21 or 22 by itsef defines ζ. The ine through the focus and parae to the directrix is defined by θ = π/2. By 21 or from the origina definition of ζ, this ine meets ζ in two points, L 0 and L 1, whose coordinates are ec, π/2 and ec, π/2. It wi be convenient to denote the distance L 0 L 1 by 2: this means defining = ec. 23 Then 18, 21 and 22 can be rewritten as r = er cos θ +, 24 r = 1 e cos θ 25 r = 1 e cos θ Lines through the focus. By 25, each ine θ = ϕ through the origin meets ζ in two points, namey 1 e cos ϕ, ϕ and 1 + e cos ϕ, ϕ + π, 27 uness e cos ϕ = ±1. There are three possibiities, corresponding the three kinds of conic sections: 1 If 0 < e < 1, then e cos θ is never 1, so every ine through the origin meets ζ at two points, and these points are on opposite sides of the origin; ζ is an eipse. See Figure 7. 2 If e = 1, then every ine through the origin meets ζ at two points, which are are on opposite sides of the origin, uness the ine is θ = 0: This ine meets ζ ony at /2, π, hafway between the focus and the directrix. Now ζ is a paraboa. See Figure 8.

8 8 DAVID PIERCE 1 e cos θ, θ L 0 θ V F V d L e cos θ, θ + π Figure 7. The eipse d L 0 1 e cos θ, θ V θ F 1 + e cos θ, θ + π L 1 Figure 8. The paraboa 3 Suppose e > 1. then cos α = 1/e for some α such that 0 < α < π/2. If α < ϕ < 2π α, then the ine θ = ϕ meets ζ at two points, on opposite sides of the origin, as in the eipse and paraboa. If α < ϕ < α, then the ine θ = ϕ meets ζ at two points, on the same side of the origin. Each of the ines θ = α and θ = α meets ζ once, at /2, π + α or /2, π α. Here ζ is an hyperboa. It is reay two curves: ζ 0, given by 25, where α < θ < 2π α; ζ 1, given by 25, where α < θ < α; or by 26, where π α < θ < π + α. See Figure Distances. The ine through the focus F perpendicuar to the directrix d is the axis of ζ. Then ζ is symmetric about its axis, because of the origina definition, or by 25. A point of ζ that ies on the axis is a vertex of ζ. Again, there are three cases: 1 Say 0 < e < 1, so ζ is an eipse. Then ζ has a vertex V, with coordinates /1 + e, π, and a vertex V, given by /1 e, 0. Since we have 0 < 1 e 1 e cos θ 1 + e, e 1 e cos θ 1 e. 29

9 CONIC SECTIONS 9 1 e cos ϕ, ϕ L 0 1 e cos ψ, ψ ζ 0 e cos θ = 1 ζ 1 V V d 1 + e cos ϕ, ϕ + π ψ L 1 ϕ e cos θ = 1 Figure 9. The hyperboa By 25 then, V is the point of ζ that is cosest to the focus, and V is the point furthest from F. Aso, See Figure 10. V V = 1 + e + 1 e = 2 1 e V F V V F Figure 10. Extreme points in the eipse and paraboa 2 Say e = 1, so ζ is a paraboa. Then it has a unique vertex, V, with coordinates /2, π. As in the case of the eipse, so in the paraboa, V is the point of ζ cosest to the focus; but there is no furthest point. Again, see Figure Say e > 1, so ζ is an hyperboa. Then it has two vertices, V and V, with coordinates /e+1, π and /e 1, π respectivey. As before, suppose cos α =

10 10 DAVID PIERCE 1/e, where 0 < α < π/2. If α < θ < α, then 1 < cos θ 1, e 31 1 < e cos θ e, 32 0 < e cos θ 1 e 1, 33 0 < e 1 e cos θ 1 ; 34 so V is the point of ζ 1 cosest to the focus. If α < θ < 2π α, then 1 cos θ < 1 e, 35 e e cos θ < 1, 36 1 < e cos θ e, 37 0 < 1 e cos θ e + 1, 38 e + 1 so V is the point of ζ 0 cosest to the focus. Finay, See Figure 11. V V = e 1 1 e cos θ ; 39 e + 1 = 2 e V V F Figure 11. Extreme points in the hyperboa In both the eipse and the hyperboa then, the distance between the two vertices is 2/ e 2 1 ; this may aso be denoted by 2a, so that a = e

11 CONIC SECTIONS Areas. Let P be an arbitrary point with coordinates r, θ on ζ, and et the foot of the perpendicuar from P to the axis of ζ be Q as in Figure 6. Then Q has coordinates r cos θ, 0. We consider the position of Q with respect to the vertices: 1 If 0 < e < 1, then by 29 and 24 r 1 + e 1 e, 42 er cos θ + 1 e, e e 1 + e 1 + e so Q is between V and V, and 2 If e = 1, then er cos θ e 1 e, 44 r cos θ 1 e ; 45 V Q = r cos θ e, 46 V Q = r cos θ e 2 r = r cos θ +, 48 2 r cos θ, 49 V Q = r cos θ If e > 1, then there are two cases: a if P is on ζ 0, then 1 + e r = er cos θ +, 51 e er cos θ, 1 + e 52 r cos θ, 1 + e 53 V Q = r cos θ + e + 1, 54 V Q = r cos θ + e 1 ; 55

12 12 DAVID PIERCE b if P is on ζ 1, then r = er cos θ +, e 1 56 e er cos θ, e 1 57 r cos θ, 58 e 1 V Q = r cos θ +, 59 e + 1 V Q = r cos θ + In either case, Q is not between V and V. Now we can compute: e P Q 2 = r 2 sin 2 θ 61 There are two cases: If e = 1, then this equation becomes If e 1, then = r 2 r 2 cos 2 θ 62 = er cos θ + 2 r 2 cos 2 θ 63 = r[e + 1] cos θ + r[e 1] cos θ P Q 2 = 2r cos θ + = 2 V Q. 65 P Q 2 = e 2 1 r cos θ + r cos θ + e + 1 e 1 66 = e 2 1 V Q V Q 67 = 2 V Q V Q. 68 V V Let V R be drawn perpendicuar to the axis of ζ so that V R = 2. This ine segment is caed the atus rectum of ζ. This is the term commony used in Engish, athough it is the Latin transation of the origina Greek found in Apoonius; however, the itera Engish transation, upright side, is used in [2]. Then the square with side P Q is the area of the rectange with sides V Q and V R, if ζ is a paraboa; fas short of this area, if ζ is an eipse; exceeds this area, if ζ is an hyperboa. This is what is suggested by the Greek names of the curves. See Figure The rectanguar equations. For the paraboa, choose a rectanguar coordinate system in which V is the origin and the X-axis is the axis of ζ. Then 65 becomes y 2 = 2x. 69 This is the standard rectanguar equation for a paraboa. The focus is at /2, 0, and the directrix is given by x + /2 = 0.

13 P CONIC SECTIONS 13 P V R Q V V Q R P P V V Q Q V V R R Figure 12. Conic sections as determined by equations of areas For the eipse and the hyperboa, et the origin of a rectanguar coordinate system be the midpoint O of V V : this is the center of the conic section. Let the X-axis contain the vertices. Then the vertices wi have coordinates ±a, 0. By 67, the curve is symmetric about the new Y -axis. In particuar, the curve has, not just one focus, but two foci; hence it has, not just one directrix, but two directrices, one for each focus. The curve is now given by y 2 = e 2 1 x a x + a = e 2 1 x 2 a Moreover, by the previous subsection, in the eipse, e 2 1 and x 2 a 2 are both negative; in the hyperboa, positive. Hence 70 can be written Recaing 41, we can write y 2 = e 2 1x 2 a 2, 71 y 2 a 2 e 2 1 = x2 1, a2 72 x 2 a + y 2 = 1. 2 a 2 1 e 2 73 x 2 a ± y2 = 1, 74 2 a where the upper sign is for the eipse, and the ower is for the hyperboa. We may et b be the positive number such that b 2 = a, 75 so that 74 becomes x 2 a ± y2 = b2 The Y -intercepts of the eipse are 0, ±b; the hyperboa has no Y -intercepts. By 41 and 75, e = 1 b2 a ; 77 2

14 14 DAVID PIERCE where again the upper sign is for the eipse. Aso, F O = a 1 + e = a a1 e2 = a a1 e = ae; e so the foci are at ±ae, 0. Likewise, do = ae ± e = ae + a1 e2 = a e e ; 79 so the directrices are given by x ± a/e = 0. Finay, the hyperboa given by 76 does not meet the two ines given by x 2 a y2 = b2 These ines given aso by ay ± bx = 0 are the asymptotes of the hyperboa. Their sopes are ±b/a. In genera, a ine through O meets the hyperboa if and ony if the sope of the ine is ess than b/a in absoute vaue. Indeed, the equations y = mx, x 2 a y2 2 b = 1, 81 2 if soved simutaneousy, yied x 2 a m2 x 2 = 1, 82 2 b 2 b 2 a 2 m2 = b2 x, 2 83 b 2 a 2 m2 > 0, 84 m 2 < b2 a 2, 85 m < b a ; 86 and if the ast inequaity hods, then there is a simutaneous soution, obtainabe from 83 and then 81. The two hyperboas x 2 /a 2 y 2 /b 2 = ±1 have the same asymptotes. Aso, their foci are at the same distance from the center, namey a 2 + b 2. Such hyperboas are conjugate. The eipse x 2 /a 2 + y 2 /b 2 = 1 is tangent to them at their vertices. See Figure 13. The segment joining the two vertices of an eipse is the major axis of the eipse; the minor axis passes through the center, but is perpendicuar to the major axis. A circe can be described as an eipse of eccentricity 0. Stricty, however, a circe is not a conic section by the definition given in 3.1. The circe does not have a directrix. However, the circe is a kind of imit of the eipses with the same focus and atus rectum, as the directrix moves indefinitey far away which means the eccentricity tends to 0. See Figure 14. References [1] Apoonius of Perga. Apoonius of Perga: Treatise on Conic Sections. University Press, Cambridge, UK, Edited by T. L. Heath in modern notation, with introductions incuding an essay on the earier history of the subject.

15 CONIC SECTIONS 15 Figure 13. Conjugate hyperboas [2] Apoonius of Perga. On conic sections. Great Books of the Western Word, no. 11. Encycopaedia Britannica, Inc., Chicago, London, Toronto, [3] Apoonius of Perga. Conics. Books I III. Green Lion Press, Santa Fe, NM, revised edition, Transated and with a note and an appendix by R. Catesby Taiaferro, With a preface by Dana Densmore and Wiiam H. Donahue, an introduction by Harvey Faumenhaft, and diagrams by Donahue, Edited by Densmore. [4] Archimedes. The works of Archimedes. Vo. I. Cambridge University Press, Cambridge, The two books on the sphere and the cyinder, Transated into Engish, together with Eutocius commentaries, with commentary, and critica edition of the diagrams by Revie Netz. [5] Car B. Boyer. A history of mathematics. John Wiey & Sons Inc., New York, [6] Juian Lowe Cooidge. A history of the conic sections and quadric surfaces. Dover Pubications Inc., New York, [7] René Descartes. The Geometry of René Descartes. Dover Pubications, Inc., New York, Transated from the French and Latin by David Eugene Smith and Marcia L. Latham, with a facsimie of the first edition of [8] Isaac Newton. The Mathematica Principes of Natura Phiosophy [9] Lucio Russo. The forgotten revoution. Springer-Verag, Berin, How science was born in 300 BC and why it had to be reborn, Transated from the 1996 Itaian origina by Sivio Levy. [10] Ivor Thomas, editor. Seections iustrating the history of Greek mathematics. Vo. I. From Thaes to Eucid. Harvard University Press, Cambridge, Mass., With an Engish transation by the editor. [11] Ivor Thomas, editor. Seections iustrating the history of Greek mathematics. Vo. II. From Aristarchus to Pappus. Harvard University Press, Cambridge, Mass, With an Engish transation by the editor.

16 16 DAVID PIERCE Figure 14. The circe as a imit of conics Mathematics Department, Midde East Technica University, Ankara 06531, Turkey E-mai address: dpierce@metu.edu.tr URL:

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