Pg 1 of 77 AGI CONICS Jim Wright

Pg 2 of 77 AGI CONICS Cones (Menaechmus & Appollonius) Menaechmus 0350 BC Plato’s student Appollonius BC Eight Books on Conics Kepler Kepler’s Laws Pascal Pascal’s Theorem Newton Newton’s Laws to Conic LaGrange Propagate Pos & Vel Conic Brianchon Brianchon’s Theorem Dandelin From Theorem to Definition Variation of Parameters Orbits of Binary Stars

Pg 3 of 77 AGI PARABOLA

Pg 4 of 77 AGI ELLIPSE

Pg 5 of 77 AGI Cone Flat Pattern for Ellipse

Pg 6 of 77 AGI Conic Factory

Pg 7 of 77 AGI 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

Pg 8 of 77 AGI Dandelin’s Cone-Sphere Proof Ellipse

Pg 9 of 77 AGI Sphere Tangents P F1F1 C PF 1 = PC

Pg 10 of 77 AGI 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

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

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

Pg 13 of 77 AGI 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

Pg 14 of 77 AGI 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

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

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

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

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

Pg 19 of 77 AGI 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?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Pg 38 of 77 AGI 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

Pg 39 of 77 AGI 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?

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

Pg 41 of 77 AGI 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

Pg 42 of 77 AGI 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

Pg 43 of 77 AGI 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

Pg 44 of 77 AGI 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

Pg 45 of 77 AGI 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

Pg 46 of 77 AGI 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

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

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

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

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

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

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

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

Pg 54 of 77 AGI a b c d e f

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

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

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

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

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

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

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

Pg 62 of 77 AGI Brianchon’s Theorem

Pg 63 of 77 AGI Dandelin 1825 Cones and Spheres

Pg 64 of 77 AGI Conic Factory

Pg 65 of 77 AGI 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

Pg 66 of 77 AGI Dandelin’s Cone-Sphere Theorem Ellipse

Pg 67 of 77 AGI Dandelin’s Cone-Sphere Theorem Hyperbola

Pg 68 of 77 AGI Dandelin’s Cone-Sphere Theorem Parabola

Pg 69 of 77 AGI 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)

Pg 70 of 77 AGI 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

Pg 71 of 77 AGI 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

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

Pg 73 of 77 AGI 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)

Pg 74 of 77 AGI 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

Pg 75 of 77 AGI CONICS Conic Factory (Menaechmus & Appollonius) Menaechmus 350 BC Plato’s student Appollonius BC Eight Books on Conics Kepler Kepler’s Laws Pascal Pascal’s Theorem Newton Newton’s Laws to Conic Brianchon Brianchon’s Theorem Dandelin From Theorem to Definition Variation of Parameters (VOP) Orbits of Binary Stars

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

Pg 77 of 77 AGI Questions?