On the
Geometry
of
Orbits
The Possible Orbits
The Possible Orbits
circle
The Possible Orbits
ellipse
The Possible Orbits
parabola
The Possible Orbits
hyperbola
Speed and Distance
4000 mi
17,600 mph
1.4 hr
Speed and Distance
3,500 mph
26,200 mi
Add 32%
23,200 mph
10.4 hr
Speed and Distance
120,000 mi
Add 39%
24,500 mph
Speed and Distance
240,000 mi
?
Speed and Distance
240,000 mi
Add 40%
24,640 mph
Speed and Distance
“infinite
ellipse”
Add 41.4%
24,900 mph
Speed and Distance
parabola
“escape speed”
24,900 mph
Speed and Distance
hyperbola
more than
escape speed
Speed and Distance
parabola
terminal
velocity:
speed
“escape speed”
24,900 mph
0
Speed and Distance
hyperbola
terminal
velocity:
speed
excess
more than
escape speed
The
Section
TheShallow
Conic Sections
The Conic Sections
horizontal
section
The Conic Sections
shallow
section
The Conic Sections
parallel
section
The Conic Sections
steep
section
The Conic Sections
two
branches
Apollonius’s
The
Shallow
Section
Sections
of One
Cone
Apollonius’s
The Epicycle
ShallowModel
Section
Geometry of the Shallow
The Shallow
SectionSection
Geometry of the Shallow
The Shallow
SectionSection
Geometry of the Shallow
The Shallow
SectionSection
Geometry of the Shallow
The Shallow
SectionSection
F1
Geometry of the Shallow
The Shallow
SectionSection
F1
P
Geometry of the Shallow
The Shallow
SectionSection
F1
P
Tangents from a
Common Point
Geometry of the Shallow
The Shallow
SectionSection
F1
P
Geometry of the Shallow
Section
P
Geometry of the Shallow
Section
F2
P
Geometry of the Shallow
Section
F2
P
Geometry of the Shallow
Section
F2
F1
P
Geometry of the Shallow
Section
F2
Add PF1 and PF2.
F1
P
Geometry of the Shallow
Section
PF1 + PF2 =
distance between
the bands
F1
F2
P
Definition of the Ellipse
• There are two fixed points
(“foci”) for which the two
distances (“focal radii”) from
any point of the curve add up
to a fixed number.
Definition of the Ellipse
• There are two fixed points
(“foci”) for which the two
distances (“focal radii”) from
any point of the curve add up
to a fixed number.
PF1 + PF2 = constant
P
F1
F2
Properties of the Ellipse
• There are two fixed points
(“foci”) for which the two
distances (“focal radii”) from
any point of the curve add up
to a fixed number.
• The ellipse is left-right and updown symmetric.
Properties of the Ellipse
• There are two fixed points
(“foci”) for which the two
distances (“focal radii”) from
any point of the curve add up
to a fixed number.
• The main axis (the one with the
foci) is as long as the sum of
the focal radii.
Properties of the Ellipse
• There are two fixed points
(“foci”) for which the two
distances (“focal radii”) from
any point of the curve add up
to a fixed number.
• The main axis is longer than
the other:
M 2 = m2 + f 2
Properties of the Ellipse
• There are two fixed points
(“foci”) for which the two
distances (“focal radii”) from
any point of the curve add up
to a fixed number.
• The ratio = f/M (the
“eccentricity”) determines the
shape of the ellipse.
Eccentricity and the Shape
of the Ellipse
M2 = m2 + f 2 and = f/M
lead to
m = M (1 –
2).
Eccentricity and the Shape
of the Ellipse
M2 = m2 + f 2 and = f/M
lead to
m = M (1 –
• Earth:
= .02
2).
m = M(.9998)
Eccentricity and the Shape
of the Ellipse
M2 = m2 + f 2 and = f/M
lead to
m = M (1 –
• Earth:
= .02
• Mars:
= .09
2).
m = M(.9998)
m = M(.996)
Eccentricity and the Shape
of Two Familiar Orbits
91
Earth
94.5
Sun
Eccentricity and the Shape
of Two Familiar Orbits
128
91
Earth
155
94.5
Sun
Mars
Definition of the Ellipse
PF1 + PF2 = constant
P
F1
F2
Definition of the Hyperbola
PF2 – PF1 = constant
F1
F2
P
Definition of the Hyperbola
• There are two fixed points
(“foci”) for which the two
distances (“focal radii”) from
any point of the curve differ by
a fixed number.
PF2 – PF1 = constant
F1
F2
P
Definition of the Hyperbola
• There are two fixed points
(“foci”) for which the two
distances (“focal radii”) from
any point of the curve differ by
a fixed number.
F1
Q
F2
QF1 – QF2 = constant
Seismography
and the Hyperbola
Suppose San Francisco
hears an earthquake at 12,
New York hears at 5,
Miami hears at 5:12.
Seismography
and the Hyperbola
distance to New York
- distance to San Francisco
= 2,000 mi
Seismography
and the Hyperbola
distance to New York
- distance to San Francisco
= 2,000 mi
Seismography
and the Hyperbola
distance to Miami
- distance to San Francisco
= 2,200 mi
Seismography
and the Hyperbola
Location:
Elko NV
More Geometry
The of
Shallow
Section
the Sections
F1
P
More Geometry
The of
Shallow
Section
the Sections
F1
P
More Geometry
The of
Shallow
Section
the Sections
F1
Q
P
More Geometry
The of
Shallow
Section
the Sections
P
F1
35
Q
65
More Geometry
The of
Shallow
Section
the Sections
P
F1
35
Q
RS
65
More Geometry
The of
Shallow
Section
the Sections
P
35
Q
PS/PQ=
sin 35
S
More Geometry
The of
Shallow
Section
the Sections
P
PS/PR=
sin 65
RS
65
More Geometry
The of
Shallow
Section
the Sections
P
F1
PR/PF1=
sin 65
RS
65
More Geometry
The of
Shallow
Section
the Sections
F1
Q
PF1/PQ=
sin 35 /sin 65
P
More Geometry
of the Sections
PF1/PQ =
sin 35 /sin 65
P
F1
Q
More Geometry
of the Sections
PF1/PQ =
constant less
than 1
P
F1
Q
More Geometry
of the Sections
PF1/PQ =
eccentricity
P
F1
Q
Alternate Description
of the Ellipse
There is a line
(“directrix”) such that
distance to focus
distance to line =
eccentricity
P
F1
Q
Eccentricity in
the Sections
35
eccentricity =
sin 35 /sin 65
65
Eccentricity in
the Sections
0
eccentricity =
sin 0 /sin 65
Eccentricity in
the Sections
0
eccentricity = 0
Eccentricity in
the Sections
The “eccentricity”
of the circle is 0.
Eccentricity in
the Sections
35
eccentricity =
sin 35 /sin 65
65
Eccentricity in
the Sections
65
eccentricity =
sin 65 /sin 65
65
Eccentricity in
the Sections
eccentricity = 1
Eccentricity in
the Sections
The “eccentricity” of
the parabola is 1.
Definition of the
Parabola
PF1/PQ =
sin 65 /sin 65
P
F1
Q
Definition of the
Parabola
PF1 = PQ
P
F1
Q
Definition of the
Parabola
P
F1
For every
point, distance
to the focus
Q
equals
distance to the
directrix.
Eccentricity in
the Sections
80
eccentricity =
sin 80 /sin 65
65
Eccentricity in
the Sections
PF1/PQ =
sin 80 /sin 65
F1
P
Q
Eccentricity in
the Sections
PF1/PQ =
constant
greater than 1
F1
P
Q
Geometry of the
Steep Section
“Eccentricity”
of the
hyperbola
exceeds 1.
F1
P
Q
Speed and Eccentricity
17,600 mph
Speed and Eccentricity
circle
eccentricity
= (v/v0)2 – 1
= 12 – 1
=0
17,600 mph
Speed and Eccentricity
26,200 mi
Add 32%
23,200 mph
Speed and Eccentricity
ellipse
eccentricity
= (v/v0)2 – 1
= 1.322 – 1
0.74
Add 32%
23,200 mph
26,200 mi
Speed and Eccentricity
ellipse
eccentricity
= (v/v0)2 – 1
= 1.392 – 1
0.93
Add 39%
24,500 mph
120,000 mi
Speed and Eccentricity
eccentricity
= (v/v0)2 – 1
= 1.4142 – 1
Add 41.4%
24,900 mph
Speed and Eccentricity
parabola
eccentricity
= (v/v0)2 – 1
= ( 2)2 – 1
=1
Add 41.4%
24,900 mph
Speed and Eccentricity
hyperbola
eccentricity
= (v/v0)2 – 1
= 1.52 – 1
= 1.25
Add 50%
26,400 mph
Elements of
the Parabola
one
axis
F
one directrix
Elements of
the Parabola
“baseline”
F
Extent of
the Parabola
F
no points below the
baseline
Elements of
the Parabola
no points
along the
axis
F
no points below the
baseline
Extent of
the Parabola
points in
all other
directions
F
Elements of
the Hyperbola
F1
one axis
Elements of
the Hyperbola
F1
F2
second focus
and directrix
Elements of
the Hyperbola
F1
second axis
F2
Elements of
the Hyperbola
F1
F2
Elements of
the Hyperbola
F1
F2
Extent of
the Hyperbola
Hyperbola is
confined to the
gray region.
F1
F2
Reflection Properties:
the Ellipse
F1
F2
Reflection Properties:
the Ellipse
F1
F2
Reflection Properties:
the Ellipse
F1
F2
Reflection Properties:
the Parabola
P
F
Reflection Properties:
the Parabola
P
F
Reflection Properties:
the Hyperbola
F1
F2
Reflection Properties:
the Hyperbola
F1
F2
Reflection Properties:
the Hyperbola
F1
F2
Reflection Properties:
the Hyperbola
F1
F2
Telescopes and the Conics
Telescopes and the Conics
Telescopes and the Conics
Telescopes and the Conics
Telescopes and the Conics
Telescopes and the Conics