Transcript AGI conference presentation

CONICS
Jim Wright
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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
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PARABOLA
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ELLIPSE
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Cone Flat Pattern for Ellipse
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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
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Dandelin’s Cone-Sphere Proof
Ellipse
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Sphere Tangents
P
F1
C
PF1 = PC
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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
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Conics without Cones
• How to construct a conic with
pencil and straight-edge
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PASCAL’S THEOREM
1640
Pairs of opposite sides of a hexagon inscribed in a
conic intersect on a straight line
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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
A
B
F
C
BC-EF
CD-FA
D
E
PASCAL
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Hexagon
ABCDEF(A)
PASCAL
B
Opposite Sides
AB-DE
BC-EF
F
D
CD-FA
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 Pascal’s Theorem, straightedge and pencil
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Hexagon
ABCDEX(A)
PASCAL
B
Opposite Sides
AB-DE
BC-EX
CD-XA
D
C
A
E
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Hexagon
ABCDEX(A)
B
Opposite Sides
AB-DE
BC-EX
CD-XA
D
C
A
E
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Hexagon
ABCDEX(A)
B
Opposite Sides
AB-DE
BC-EX
CD-XA
D
P1
C
A
E
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Hexagon
ABCDEX(A)
Opposite Sides
B
XA
AB-DE
BC-EX
CD-XA
D
P1
C
A
E
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Hexagon
ABCDEX(A)
B
Opposite Sides
XA
AB-DE
BC-EX
CD-XA
D
P1
P2
C
A
E
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Hexagon
ABCDEX(A)
B
Opposite Sides
XA
AB-DE
BC-EX
CD-XA
D
P1
P2
Pascal Line
C
A
E
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Hexagon
ABCDEX(A)
Opposite Sides
B
XA
AB-DE
BC-EX
CD-XA
D
P1
P2
P3
Pascal Line
C
A
E
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Hexagon
ABCDEX(A)
B
Opposite Sides
XA
AB-DE
X
BC-EX
CD-XA
D
P1
P2
P3
Pascal Line
C
A
E
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Hexagon
ABCDEX(A)
B
Opposite Sides
AB-DE
X
BC-EX
CD-XA
D
P1
C
A
P2
P3
E
q
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Hexagon
ABCDEX(A)
B
Opposite Sides
EX
AB-DE
BC-EX
CD-XA
D
P1
C
A
P2
P3
E
q
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X
EX
Hexagon
ABCDEX(A)
B
Opposite Sides
AB-DE
BC-EX
CD-XA
D
P1
C
A
P2
P3
E
q
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X
EX
Hexagon
ABCDEX(A)
B
Opposite Sides
AB-DE
BC-EX
CD-XA
P2
D
P1
C
A
P3
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X
EX
Hexagon
ABCDEX(A)
B
Opposite Sides
AB-DE
BC-EX
CD-XA
P2
D
P1
Pascal Line
C
A
P3
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Hexagon
ABCDEX(A)
Pascal
1623 – 1662
EX
Brianchon 1785 - 1864
B
Opposite Sides
AB-DE
BC-EX
CD-XA
D
P1
P2
P3
Pascal Line
C
A
E
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Hexagon
ABCDEX(A)
Pascal
1623 – 1662
EX
Brianchon 1785 - 1864
B
Opposite Sides
AB-DE
BC-EX
CD-XA
D
P1
P2
P3
Pascal Line
C
A
E
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Hexagon
ABCDEX(A)
EX
B
Opposite Sides
X
AB-DE
BC-EX
CD-XA
D
P1
P2
P3
Pascal Line
C
A
E
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Pascal’s Theorem
1640
EX
Brianchon’s Theorem 1806
B
D
C
A
E
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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
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Hexagon
abcdef
Lines ab.de, bc.ef, and cd.fa are concurrent
How many lines are required to uniquely specify a conic?
c
Opposite
Vertices
ab.de
b
bc.ef
d
cd.fa
e
a
f
Brianchon’s Theorem
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Line Conic Curve
• Conic defined uniquely by 5 lines
• Add more lines with Brianchon’s Theorem
(straight-edge and pencil)
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Hexagon
axcdef
Brianchon’s Theorem
Opposite
Vertices
ax.de
xc.ef
cd.fa
c
d
e
a
f
Lines ax.de, xc.ef, and cd.fa are concurrent
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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
AGI
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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
AGI
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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
AGI
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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
AGI
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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
AGI
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Hexagon
axcdef
Opposite
Vertices
ax.de
xc.ef
cd.fa
c
x
d
e
a
f
AGI
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Brianchon’s Theorem
a
Hexagon
abcdex
b
Opposite
Vertices
ab.de
bc.ex
cd.xa
c
d
e
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a
Hexagon
abcdex
Opposite
Vertices
ab.de
bc.ex
cd.xa
b
c
d
e
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Hexagon
abcdex
a
Opposite
Vertices
ab.de
bc.ex
cd.xa
b
c
ex
d
e
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Hexagon
abcdex
a
Opposite
Vertices
ab.de
bc.ex
cd.xa
b
c
ex
d
e
AGI
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Hexagon
abcdex
a
xa
b
Opposite
Vertices
ab.de
bc.ex
cd.xa
c
ex
d
e
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Hexagon
abcdex
a
Opposite
Vertices
ab.de
bc.ex
cd.xa
xa
b
x
c
ex
d
e
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a
b
f
c
d
e
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Change the Hexagon
a
Hexagon
abcdxe
b
f
Opposite
Vertices
ab.dx
bc.xe
cd.ea
c
d
e
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a
Hexagon
abcdxe
b
f
Opposite
Vertices
ab.dx
bc.xe
cd.ea
c
d
e
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a
Hexagon
abcdxe
b
f
Opposite
Vertices
ab.dx
bc.xe
cd.ea
c
d
e
AGI
dx
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a
Hexagon
abcdxe
b
f
Opposite
Vertices
ab.dx
bc.xe
cd.ea
c
d
e
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dx
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ab.dx
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a
Hexagon
abcdxe
b
f
Opposite
Vertices
ab.dx
bc.xe
cd.ea
c
d
bc.xe
e
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ab.dx
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a
Hexagon
abcdxe
b
f
Opposite
Vertices
ab.dx
bc.xe
cd.ea
c
d
bc.xe
e
AGI
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ab.dx
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Hexagon
abcdxe
a
Opposite
Vertices
ab.dx
bc.xe
cd.ea
b
f
c
g
d
e
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Brianchon’s Theorem
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Dandelin
1825
Cones and Spheres
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Conic Factory
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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
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Dandelin’s Cone-Sphere Theorem
Ellipse
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Dandelin’s Cone-Sphere Theorem
Hyperbola
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Dandelin’s Cone-Sphere Theorem
Parabola
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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)
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x2
+
y2
=
x = (p – r)/e
p/e = x + r/e
y = r sin v
sin v = y/r
r2
Y axis
Dandelin’s Conic: p = r (1 + e cos v)
Kepler’s First Law
p/e
S
p
r/e
r
y
p = q (1 + e), when r = q
v x
F
q/e
X axis
q
directrix
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Dandelin’s Conic: p = r (1 + e cos v)
Kepler’s First Law
Semi-major axis: a = q/(1 - e), for e ≠ 1
AGI
Parabola:
e=1
and a is undefined
Ellipse:
0 ≤ e < 1 and a > 0
Hyperbola:
e>1
and a < 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, Ω
• Orient the ellipse within the plane with ω
• Earth orbit at time t0 is defined by these 6 constants
• Earth orbit at time t1 > 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
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
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STK, Astrogator, ODTK
Extensive use of all three conics and VOP
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Questions?
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