1 / 77

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

analu
Download Presentation

CONICS

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related