CONICS

1
CONICS

• Jim Wright

2
CONICS

• Cones (Menaechmus Appollonius)
• Menaechmus 0350 BC Platos student
• Appollonius 0262-0200 BC Eight Books on
  Conics
• Kepler 1571-1630 Keplers Laws
• Pascal 1623-1662 Pascals Theorem
• Newton 1642-1727 Newtons Laws to
  Conic
• LaGrange 1736-1813 Propagate Pos Vel
  Conic
• Brianchon 1785-1864 Brianchons 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
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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
Dandelins Cone-Sphere ProofEllipse
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Sphere Tangents
P
F1
C
PF1 PC
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Dandelins 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

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Conics without Cones

• How to construct a conic with pencil and
  straight-edge

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PASCALS 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

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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

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

• Point Conic defined uniquely by 5 points
• Add more points with Pascals Theorem,
  straight-edge and pencil

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Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-X
A
PASCAL
B
D
C
A
E
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Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-X
A
B
D
C
A
E
23
Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-X
A
B
D
P1
C
A
E
24
Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-X
A
B
XA
D
P1
C
A
E
25
Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-X
A
B
XA
D
P1
P2
C
A
E
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Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-X
A
B
XA
D
P1
P2
Pascal Line
C
A
E
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Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-X
A
B
XA
D
P1
P2
P3
Pascal Line
C
A
E
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Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-X
A
B
XA
X
D
P1
P2
P3
Pascal Line
C
A
E
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Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-X
A
B
X
D
P1
C
A
P2
P3
E
q
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Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-X
A
EX
B
D
P1
C
A
P2
P3
E
q
31
X
Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-X
A
EX
B
D
P1
C
A
P2
P3
E
q
32
X
Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-X
A
EX
B
P2
D
P1
C
A
P3
E
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X
Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-X
A
EX
B
P2
D
P1
Pascal Line
C
A
P3
E
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Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-X
A
Pascal 1623 1662 Brianchon 1785 - 1864
EX
B
P2
D
P3
P1
Pascal Line
C
A
E
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Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-X
A
Pascal 1623 1662 Brianchon 1785 - 1864
EX
B
P2
D
P3
P1
Pascal Line
C
A
E
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Hexagon ABCDEX(A) Opposite Sides AB-DE BC-EX CD-X
A
EX
B
X
P2
D
P3
P1
Pascal Line
C
A
E
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Pascals Theorem 1640 Brianchons Theorem
1806
EX
B
D
C
A
E
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Brianchons 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 Pascals Theorem
• Discovered 166 years after Pascals Theorem

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Hexagon abcdef Opposite Vertices ab.de bc.ef cd.f
a
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
Brianchons Theorem
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Line Conic Curve

• Conic defined uniquely by 5 lines
• Add more lines with Brianchons Theorem
  (straight-edge and pencil)

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Hexagon axcdef Opposite Vertices ax.de xc.ef cd.f
a
Brianchons 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.f
a
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.f
a
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.f
a
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.f
a
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.f
a
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.f
a
c
x
d
e
a
f
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Brianchons Theorem
a
Hexagon abcdex Opposite Vertices ab.de bc.ex cd.x
a
b
c
d
e
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a
Hexagon abcdex Opposite Vertices ab.de bc.ex cd.x
a
b
c
d
e
50
Hexagon abcdex Opposite Vertices ab.de bc.ex cd.x
a
a
b
c
ex
d
e
51
Hexagon abcdex Opposite Vertices ab.de bc.ex cd.x
a
a
b
c
ex
d
e
52
Hexagon abcdex Opposite Vertices ab.de bc.ex cd.x
a
a
xa
b
c
ex
d
e
53
Hexagon abcdex Opposite Vertices ab.de bc.ex cd.x
a
a
xa
b
x
c
ex
d
e
54
a
b
f
c
d
e
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Change the Hexagon
a
Hexagon abcdxe Opposite Vertices ab.dx bc.xe cd.e
a
b
f
c
d
e
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a
Hexagon abcdxe Opposite Vertices ab.dx bc.xe cd.e
a
b
f
c
d
e
57
a
Hexagon abcdxe Opposite Vertices ab.dx bc.xe cd.e
a
b
f
c
d
e
dx
58
a
Hexagon abcdxe Opposite Vertices ab.dx bc.xe cd.e
a
b
f
c
d
e
dx
ab.dx
59
a
Hexagon abcdxe Opposite Vertices ab.dx bc.xe cd.e
a
b
f
c
d
bc.xe
e
dx
ab.dx
60
a
Hexagon abcdxe Opposite Vertices ab.dx bc.xe cd.e
a
b
f
c
d
bc.xe
e
dx
ab.dx
61
Hexagon abcdxe Opposite Vertices ab.dx bc.xe cd.e
a
a
b
f
c
g
d
e
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Brianchons Theorem
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Dandelin1825
Cones and Spheres
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Conic Factory
65
Dandelins 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
Dandelins Cone-Sphere TheoremEllipse
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Dandelins Cone-Sphere TheoremHyperbola
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Dandelins Cone-Sphere TheoremParabola
69
Dandelins 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
x (p r)/e p/e x r/e y r sin v
sin v y/r
x2 y2 r2 Dandelins Conic p r (1 e cos
v) Keplers First Law
Y axis
p/e
S
r/e
p
r
y
p q (1 e), when r q
v
X axis
x
q/e
q
F
directrix
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Dandelins Conic p r (1 e cos v) Keplers
First Law
Semi-major axis a q/(1 - e), for e ?
1 Parabola e 1 and a is
undefined Ellipse 0 e lt 1 and
a gt 0 Hyperbola e gt 1 and a lt 0
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Variation Of Parameters Osculating Ellipse
t2
True Trajectory
t1
Points of Osculation
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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 t0 (i.e., 6
  constants)
• Rigorously propagate the orbit in 6 osculating
  components of position and velocity (Herrick)

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Variation of Parameters (VOP)

• Ellipse in a plane is defined by a, e, and v0
  v(t0)
• Orient the plane in 3D with i, O
• Orient the ellipse within the plane with ?
• Earth orbit at time t0 is defined by these 6
  constants
• Earth orbit at time t1 gt t0 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

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CONICS

• Conic Factory (Menaechmus Appollonius)
• Menaechmus 350 BC Platos student
• Appollonius 262-200 BC Eight Books on Conics
• Kepler 1571-1630 Keplers Laws
• Pascal 1623-1662 Pascals Theorem
• Newton 1642-1727 Newtons Laws to
  Conic
• Brianchon 1785-1864 Brianchons Theorem
• Dandelin 1794-1847 From Theorem to
  Definition
• Variation of Parameters (VOP)
• Orbits of Binary Stars

76
STK, Astrogator, ODTK

• Extensive use of all three conics and VOP

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Questions?