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CONICS

CONICS. Jim Wright. 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

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CONICS

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  1. CONICS Jim Wright

  2. 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. PARABOLA

  4. ELLIPSE

  5. Cone Flat Pattern for Ellipse

  6. Conic Factory

  7. 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. Dandelin’s Cone-Sphere ProofEllipse

  9. Sphere Tangents P F1 C PF1 = PC

  10. Dandelin’s Cone-Sphere Proof • Length: PF1 = PC because both lines PF1 and PC are tangent to the same large sphere • Length: PF2 = PD because both lines PF2 and PD are tangent to the same small sphere • PC + PD is the constant distance between the two parallel circles • PC + PD = PF1 + PF2 • Then PF1 + PF2 is also constant • PF1 + PF2 constant implies ellipse with foci F1 & F2

  11. Conics without Cones • How to construct a conic with pencil and straight-edge

  12. PASCAL’S THEOREM1640 Pairs of opposite sides of a hexagon inscribed in a conic intersect on a straight line

  13. 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. 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. Hexagon ABCDEF(A) Opposite Sides AB-DE A B D E PASCAL

  16. Hexagon ABCDEF(A) Opposite Sides BC-EF B F C E PASCAL

  17. Hexagon ABCDEF(A) Opposite Sides CD-FA A F C D PASCAL

  18. Hexagon ABCDEF(A) Opposite Sides AB-DE BC-EF CD-FA A B F C D E PASCAL

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

  20. Point Conic Curve • Point Conic defined uniquely by 5 points • Add more points with Pascal’s Theorem, straight-edge and pencil

  21. Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA PASCAL B D C A E

  22. Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA B D C A E

  23. Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA B D P1 C A E

  24. Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA B XA D P1 C A E

  25. Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA B XA D P1 P2 C A E

  26. Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA B XA D P1 P2 Pascal Line C A E

  27. Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA B XA D P1 P2 P3 Pascal Line C A E

  28. Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA B XA X D P1 P2 P3 Pascal Line C A E

  29. Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA B X D P1 C A P2 P3 E q

  30. Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA EX B D P1 C A P2 P3 E q

  31. X Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA EX B D P1 C A P2 P3 E q

  32. X Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA EX B P2 D P1 C A P3 E

  33. X Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA EX B P2 D P1 Pascal Line C A P3 E

  34. Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA Pascal 1623 – 1662 Brianchon 1785 - 1864 EX B P2 D P3 P1 Pascal Line C A E

  35. Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA Pascal 1623 – 1662 Brianchon 1785 - 1864 EX B P2 D P3 P1 Pascal Line C A E

  36. Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-XA EX B X P2 D P3 P1 Pascal Line C A E

  37. Pascal’s Theorem 1640 Brianchon’s Theorem 1806 EX B D C A E

  38. Brianchon’s Theorem1806 • 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. Hexagon abcdef Opposite Vertices ab.de bc.ef cd.fa Lines ab.de, bc.ef, and cd.fa are concurrent How many lines are required to uniquely specify a conic? c b d e a f Brianchon’s Theorem

  40. Line Conic Curve • Conic defined uniquely by 5 lines • Add more lines with Brianchon’s Theorem (straight-edge and pencil)

  41. Hexagon axcdef Opposite Vertices ax.de xc.ef cd.fa Brianchon’s Theorem c d e a f Lines ax.de, xc.ef, and cd.fa are concurrent

  42. Hexagon axcdef Opposite Vertices ax.de xc.ef cd.fa c d e a f Lines ax.de, xc.ef, and cd.fa are concurrent

  43. Hexagon axcdef Opposite Vertices ax.de xc.ef cd.fa c ax d e a f Lines ax.de, xc.ef, and cd.fa are concurrent

  44. Hexagon axcdef Opposite Vertices ax.de xc.ef cd.fa c ax d e a f Lines ax.de, xc.ef, and cd.fa are concurrent

  45. Hexagon axcdef Opposite Vertices ax.de xc.ef cd.fa c ax d e a f Lines ax.de, xc.ef, and cd.fa are concurrent

  46. Hexagon axcdef Opposite Vertices ax.de xc.ef cd.fa c ax x d e a f Lines ax.de, xc.ef, and cd.fa are concurrent

  47. Hexagon axcdef Opposite Vertices ax.de xc.ef cd.fa c x d e a f

  48. Brianchon’s Theorem a Hexagon abcdex Opposite Vertices ab.de bc.ex cd.xa b c d e

  49. a Hexagon abcdex Opposite Vertices ab.de bc.ex cd.xa b c d e

  50. Hexagon abcdex Opposite Vertices ab.de bc.ex cd.xa a b c ex d e

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