1. Pg 1 of 77 AGI www.agiuc.com CONICS Jim Wright

2. Pg 2 of 77 AGI www.agiuc.com CONICS Cones (Menaechmus & Appollonius) Menaechmus 0350 BC Plato’s student Appollonius 0262-0200 BC Eight Books on Conics Kepler 1571-1630 Kepler’s Laws Pascal 1623-1662 Pascal’s Theorem Newton 1642-1727 Newton’s Laws to Conic LaGrange 1736-1813 Propagate Pos & Vel Conic Brianchon 1785-1864 Brianchon’s Theorem Dandelin 1794-1847 From Theorem to Definition Variation of Parameters Orbits of Binary Stars

3. Pg 3 of 77 AGI www.agiuc.com PARABOLA

4. Pg 4 of 77 AGI www.agiuc.com ELLIPSE

5. Pg 5 of 77 AGI www.agiuc.com Cone Flat Pattern for Ellipse

6. Pg 6 of 77 AGI www.agiuc.com Conic Factory

7. Pg 7 of 77 AGI www.agiuc.com CONIC from CONE Slice a cone with a plane See a conic in the plane Ellipse: Slice through all elements of the cone Parabola: Slice parallel to an element of cone Hyperbola: Slice through both nappes of the cone

8. Pg 8 of 77 AGI www.agiuc.com Dandelin’s Cone-Sphere Proof Ellipse

9. Pg 9 of 77 AGI www.agiuc.com Sphere Tangents P F1F1 C PF 1 = PC

10. Pg 10 of 77 AGI www.agiuc.com Dandelin’s Cone-Sphere Proof Length: PF 1 = PC because both lines PF 1 and PC are tangent to the same large sphere Length: PF 2 = PD because both lines PF 2 and PD are tangent to the same small sphere PC + PD is the constant distance between the two parallel circles PC + PD = PF 1 + PF 2 Then PF 1 + PF 2 is also constant PF 1 + PF 2 constant implies ellipse with foci F 1 & F 2

11. Pg 11 of 77 AGI www.agiuc.com Conics without Cones How to construct a conic with pencil and straight-edge

12. Pg 12 of 77 AGI www.agiuc.com PASCAL’S THEOREM 1640 Pairs of opposite sides of a hexagon inscribed in a conic intersect on a straight line

13. Pg 13 of 77 AGI www.agiuc.com Order of Hexagon Points Each distinct order of hexagon points generates a distinct hexagon Six points A, B, C, D, E, F can be ordered in 60 different ways 60 distinct Pascal lines associated with six points was called the mystic hexagram

14. Pg 14 of 77 AGI www.agiuc.com Distinct Hexagons Hexagons ABCDEF and ACBDEF are distinct and have different opposite sides ABCDEF AB.DE BC.EF CD.FA ACBDEF AC.DE CB.EF BD.FA

15. Pg 15 of 77 AGI www.agiuc.com A B D E Hexagon ABCDEF(A) Opposite Sides AB-DE PASCAL

16. Pg 16 of 77 AGI www.agiuc.com B C E F Hexagon ABCDEF(A) Opposite Sides BC-EF PASCAL

17. Pg 17 of 77 AGI www.agiuc.com A C D F Hexagon ABCDEF(A) Opposite Sides CD-FA PASCAL

18. Pg 18 of 77 AGI www.agiuc.com A B C D E F Hexagon ABCDEF(A) Opposite Sides AB-DE BC-EF CD-FA PASCAL

19. Pg 19 of 77 AGI www.agiuc.com A B C D E F Hexagon ABCDEF(A) Opposite Sides AB-DE BC-EF CD-FA PASCAL How many points are required to uniquely specify a conic?

20. Pg 20 of 77 AGI www.agiuc.com Point Conic Curve Point Conic defined uniquely by 5 points Add more points with Pascal’s Theorem, straight- edge and pencil

21. Pg 21 of 77 AGI www.agiuc.com A B C D E Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA PASCAL

22. Pg 22 of 77 AGI www.agiuc.com A B C D E Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA

23. Pg 23 of 77 AGI www.agiuc.com A B C D E Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA P1P1

24. Pg 24 of 77 AGI www.agiuc.com A B C D E Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA XA P1P1

25. Pg 25 of 77 AGI www.agiuc.com A B C D E Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA XA P1P1 P2P2

26. Pg 26 of 77 AGI www.agiuc.com A B C D E Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA XA P1P1 P2P2 Pascal Line

27. Pg 27 of 77 AGI www.agiuc.com A B C D E Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA XA P1P1 P2P2 P3P3 Pascal Line

28. Pg 28 of 77 AGI www.agiuc.com A B C D E Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA XA X P1P1 P2P2 P3P3 Pascal Line

29. Pg 29 of 77 AGI www.agiuc.com A B C D E Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA P1P1 P2P2 P3P3 q X

30. Pg 30 of 77 AGI www.agiuc.com A B C D E Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA EX P1P1 P2P2 P3P3 q

31. Pg 31 of 77 AGI www.agiuc.com A B C D E Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA EX X P1P1 P2P2 P3P3 q

32. Pg 32 of 77 AGI www.agiuc.com A B C D E Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA EX X P1P1 P2P2 P3P3

33. Pg 33 of 77 AGI www.agiuc.com A B C D E Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA EX X P1P1 P2P2 P3P3 Pascal Line

34. Pg 34 of 77 AGI www.agiuc.com A B C D E Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA EX P1P1 P2P2 P3P3 Pascal 1623 – 1662 Brianchon 1785 - 1864 Pascal Line

35. Pg 35 of 77 AGI www.agiuc.com A B C D E Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA EX P1P1 P2P2 P3P3 Pascal 1623 – 1662 Brianchon 1785 - 1864 Pascal Line

36. Pg 36 of 77 AGI www.agiuc.com A B C D E Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA EX X P1P1 P2P2 P3P3 Pascal Line

37. Pg 37 of 77 AGI www.agiuc.com A B C D E EX Pascal’s Theorem 1640 Brianchon’s Theorem 1806

38. Pg 38 of 77 AGI www.agiuc.com Brianchon’s Theorem 1806 The lines joining opposite vertices of a hexagon circumscribed about a conic are concurrent Construct a conic with tangents rather than points (straight-edge and pencil) Perfect dual to Pascal’s Theorem Discovered 166 years after Pascal’s Theorem

39. Pg 39 of 77 AGI www.agiuc.com Hexagon abcdef Opposite Vertices ab.de bc.ef cd.fa a b c d e f Brianchon’s Theorem Lines ab.de, bc.ef, and cd.fa are concurrent How many lines are required to uniquely specify a conic?

40. Pg 40 of 77 AGI www.agiuc.com Line Conic Curve Conic defined uniquely by 5 lines Add more lines with Brianchon’s Theorem (straight-edge and pencil)

41. Pg 41 of 77 AGI www.agiuc.com Hexagon axcdef Opposite Vertices ax.de xc.ef cd.fa a c d e f Brianchon’s Theorem Lines ax.de, xc.ef, and cd.fa are concurrent

42. Pg 42 of 77 AGI www.agiuc.com Hexagon axcdef Opposite Vertices ax.de xc.ef cd.fa a c d e f Lines ax.de, xc.ef, and cd.fa are concurrent

43. Pg 43 of 77 AGI www.agiuc.com Hexagon axcdef Opposite Vertices ax.de xc.ef cd.fa a c d e f Lines ax.de, xc.ef, and cd.fa are concurrent ax

44. Pg 44 of 77 AGI www.agiuc.com Hexagon axcdef Opposite Vertices ax.de xc.ef cd.fa a c d e f Lines ax.de, xc.ef, and cd.fa are concurrent ax

45. Pg 45 of 77 AGI www.agiuc.com Hexagon axcdef Opposite Vertices ax.de xc.ef cd.fa a c d e f Lines ax.de, xc.ef, and cd.fa are concurrent ax

46. Pg 46 of 77 AGI www.agiuc.com Hexagon axcdef Opposite Vertices ax.de xc.ef cd.fa a c d e f Lines ax.de, xc.ef, and cd.fa are concurrent ax x

47. Pg 47 of 77 AGI www.agiuc.com Hexagon axcdef Opposite Vertices ax.de xc.ef cd.fa a c d e f x

48. Pg 48 of 77 AGI www.agiuc.com Hexagon abcdex Opposite Vertices ab.de bc.ex cd.xa a b c d e Brianchon’s Theorem

49. Pg 49 of 77 AGI www.agiuc.com Hexagon abcdex Opposite Vertices ab.de bc.ex cd.xa a b c d e

50. Pg 50 of 77 AGI www.agiuc.com Hexagon abcdex Opposite Vertices ab.de bc.ex cd.xa a b c d e ex

51. Pg 51 of 77 AGI www.agiuc.com Hexagon abcdex Opposite Vertices ab.de bc.ex cd.xa a b c d e ex

52. Pg 52 of 77 AGI www.agiuc.com Hexagon abcdex Opposite Vertices ab.de bc.ex cd.xa a b c d e ex xa

53. Pg 53 of 77 AGI www.agiuc.com Hexagon abcdex Opposite Vertices ab.de bc.ex cd.xa a b c d e ex xa x

54. Pg 54 of 77 AGI www.agiuc.com a b c d e f

55. Pg 55 of 77 AGI www.agiuc.com Hexagon abcdxe Opposite Vertices ab.dx bc.xe cd.ea a b c d e f Change the Hexagon

56. Pg 56 of 77 AGI www.agiuc.com Hexagon abcdxe Opposite Vertices ab.dx bc.xe cd.ea a b c d e f

57. Pg 57 of 77 AGI www.agiuc.com Hexagon abcdxe Opposite Vertices ab.dx bc.xe cd.ea a b c d e f dx

58. Pg 58 of 77 AGI www.agiuc.com Hexagon abcdxe Opposite Vertices ab.dx bc.xe cd.ea a b c d e f dx ab.dx

59. Pg 59 of 77 AGI www.agiuc.com Hexagon abcdxe Opposite Vertices ab.dx bc.xe cd.ea a b c d e f dx bc.xe ab.dx

60. Pg 60 of 77 AGI www.agiuc.com Hexagon abcdxe Opposite Vertices ab.dx bc.xe cd.ea a b c d e f dx bc.xe ab.dx

61. Pg 61 of 77 AGI www.agiuc.com Hexagon abcdxe Opposite Vertices ab.dx bc.xe cd.ea a b c d e f g

62. Pg 62 of 77 AGI www.agiuc.com Brianchon’s Theorem

63. Pg 63 of 77 AGI www.agiuc.com Dandelin 1825 Cones and Spheres

64. Pg 64 of 77 AGI www.agiuc.com Conic Factory

65. Pg 65 of 77 AGI www.agiuc.com Dandelin’s Cone-Sphere Theorem Cut a conic from a right circular cone. Then the conic foci are points of contact of spheres inscribed in the cone that touch the plane of the conic

66. Pg 66 of 77 AGI www.agiuc.com Dandelin’s Cone-Sphere Theorem Ellipse

67. Pg 67 of 77 AGI www.agiuc.com Dandelin’s Cone-Sphere Theorem Hyperbola

68. Pg 68 of 77 AGI www.agiuc.com Dandelin’s Cone-Sphere Theorem Parabola

69. Pg 69 of 77 AGI www.agiuc.com Dandelin’s Conic Theorem The locus of points in a plane whose distances, r, from a fixed point (the focus, F) bear a constant ratio (eccentricity, e) to their perpendicular distances to a straight line (the directrix) Used as definition of conic (e.g., Herrick)

70. Pg 70 of 77 AGI www.agiuc.com X axis Y axis F p r x y p/e v q/e r/e q directrix S x = (p – r)/e p/e = x + r/e y = r sin v sin v = y/r x 2 + y 2 = r 2 Dandelin’s Conic: p = r (1 + e cos v) Kepler’s First Law p = q (1 + e), when r = q

71. Pg 71 of 77 AGI www.agiuc.com Dandelin’s Conic: p = r (1 + e cos v ) Kepler’s First Law Semi-major axis: a = q/(1 - e), for e ≠ 1 Parabola: e = 1 and a is undefined Ellipse: 0 ≤ e 0 Hyperbola: e > 1 and a < 0

72. Pg 72 of 77 AGI www.agiuc.com Variation Of Parameters Osculating Ellipse Points of Osculation True Trajectory t1t1 t2t2

73. Pg 73 of 77 AGI www.agiuc.com Orbit Osculates in 6 Dimensions VOP osculates in all 6 Kepler orbit element constants Transform to 6 osculating components of position and velocity, fixed at time t 0 (i.e., 6 constants) Rigorously propagate the orbit in 6 osculating components of position and velocity (Herrick)

74. Pg 74 of 77 AGI www.agiuc.com Variation of Parameters (VOP) Ellipse in a plane is defined by a, e, and v 0 = v(t 0 ) Orient the plane in 3D with i, Ω Orient the ellipse within the plane with ω Earth orbit at time t 0 is defined by these 6 constants Earth orbit at time t 1 > t 0 is defined by 6 different constants Develop a method to change the 6 constants slowly, and change one parameter v(t) fast Refer to as Variation Of Constants, also VOP

75. Pg 75 of 77 AGI www.agiuc.com CONICS Conic Factory (Menaechmus & Appollonius) Menaechmus 350 BC Plato’s student Appollonius 262-200 BC Eight Books on Conics Kepler 1571-1630 Kepler’s Laws Pascal 1623-1662 Pascal’s Theorem Newton 1642-1727 Newton’s Laws to Conic Brianchon 1785-1864 Brianchon’s Theorem Dandelin 1794-1847 From Theorem to Definition Variation of Parameters (VOP) Orbits of Binary Stars

76. Pg 76 of 77 AGI www.agiuc.com STK, Astrogator, ODTK Extensive use of all three conics and VOP

77. Pg 77 of 77 AGI www.agiuc.com Questions?