On the. Geometry. of Orbits
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1 On the Geometry of Orbits
2 The Possible Orbits
3 The Possible Orbits circle
4 The Possible Orbits ellipse
5 The Possible Orbits parabola
6 The Possible Orbits hyperbola
7 Speed and Distance 4000 mi 17,600 mph 1.4 hr
8 Speed and Distance 3,500 mph 26,200 mi Add 32% 23,200 mph 10.4 hr
9 Speed and Distance 120,000 mi Add 39% 24,500 mph
10 Speed and Distance 240,000 mi?
11 Speed and Distance 240,000 mi Add 40% 24,640 mph
12 Speed and Distance infinite ellipse Add 41.4% 24,900 mph
13 Speed and Distance parabola escape speed 24,900 mph
14 Speed and Distance hyperbola more than escape speed
15 Speed and Distance parabola terminal velocity: speed 0 escape speed 24,900 mph
16 Speed and Distance hyperbola terminal velocity: speed excess more than escape speed
17 The Shallow Section The Conic Sections
18 The Conic Sections horizontal section
19 The Conic Sections shallow section
20 The Conic Sections parallel section
21 The Conic Sections steep section
22 The Conic Sections two branches
23 The Shallow Section Apollonius s Sections of One Cone
24 The Shallow Section Apollonius s Epicycle Model
25 The Shallow Section Geometry of the Shallow Section
26 The Shallow Section Geometry of the Shallow Section
27 The Shallow Section Geometry of the Shallow Section
28 Geometry of the Shallow The Shallow Section Section F 1
29 Geometry of the Shallow The Shallow Section Section F 1 P
30 Geometry of the Shallow The Shallow Section Section F 1 P
31 Tangents from a Common Point
32 Geometry of the Shallow The Shallow Section Section F 1 P
33 Geometry of the Shallow Section P
34 Geometry of the Shallow Section F 2 P
35 Geometry of the Shallow Section F 2 P
36 Geometry of the Shallow Section F 2 F 1 P
37 Geometry of the Shallow Section Add PF 1 and PF 2. F 2 F 1 P
38 Geometry of the Shallow Section PF 1 + PF 2 = distance between the bands F 1 P F 2
39 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.
40 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. PF 1 + PF 2 = constant P F 1 F 2
41
42 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.
43 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.
44 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 = m 2 + f 2
45 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.
46 Eccentricity and the Shape of the Ellipse M 2 = m 2 + f 2 and = f/m lead to m = M (1 2 ).
47 Eccentricity and the Shape of the Ellipse M 2 = m 2 + f 2 and = f/m lead to m = M (1 2 ). Earth: =.02 m = M(.9998)
48 Eccentricity and the Shape of the Ellipse M 2 = m 2 + f 2 and = f/m lead to m = M (1 2 ). Earth: =.02 m = M(.9998) Mars: =.09 m = M(.996)
49 Eccentricity and the Shape of Two Familiar Orbits 91 Earth Sun 94.5
50 Eccentricity and the Shape of Two Familiar Orbits 128 Mars 91 Earth Sun
51 Definition of the Ellipse PF 1 + PF 2 = constant P F 1 F 2
52 Definition of the Hyperbola PF 2 PF 1 = constant F 1 P F 2
53 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. PF 2 PF 1 = constant F 1 P F 2
54 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. F 1 F 2 Q QF 1 QF 2 = constant
55 Seismography and the Hyperbola Suppose San Francisco hears an earthquake at 12, New York hears at 5, Miami hears at 5:12.
56 Seismography and the Hyperbola distance to New York - distance to San Francisco = 2,000 mi
57 Seismography and the Hyperbola distance to New York - distance to San Francisco = 2,000 mi
58 Seismography and the Hyperbola distance to Miami - distance to San Francisco = 2,200 mi
59 Seismography and the Hyperbola Location: Elko NV
60 More Geometry The Shallow Section of the Sections F 1 P
61 More Geometry The Shallow Section of the Sections F 1 P
62 More Geometry The Shallow Section of the Sections F 1 P Q
63 More Geometry The Shallow Section of the Sections F 1 P Q 35 65
64 More Geometry The Shallow Section of the Sections F 1 P Q R S 35 65
65 More Geometry The Shallow Section of the Sections P 35 Q PS/PQ= sin 35 S
66 More Geometry The Shallow Section of the Sections P PS/PR= sin 65 R 65 S
67 More Geometry The Shallow Section of the Sections F 1 P PR/PF 1 = sin 65 R 65 S
68 More Geometry The Shallow Section of the Sections F 1 P Q PF 1 /PQ= sin 35 /sin 65
69 More Geometry of the Sections PF 1 /PQ = sin 35 /sin 65 P F 1 Q
70 More Geometry of the Sections PF 1 /PQ = constant less than 1 P F 1 Q
71 More Geometry of the Sections PF 1 /PQ = eccentricity P F 1 Q
72 Alternate Description of the Ellipse There is a line ( directrix ) such that distance to focus distance to line = eccentricity P F 1 Q
73 Eccentricity in the Sections 35 eccentricity = sin 35 /sin 65 65
74 Eccentricity in the Sections 0 eccentricity = sin 0 /sin 65
75 Eccentricity in the Sections 0 eccentricity = 0
76 Eccentricity in the Sections The eccentricity of the circle is 0.
77 Eccentricity in the Sections 35 eccentricity = sin 35 /sin 65 65
78 Eccentricity in the Sections 65 eccentricity = sin 65 /sin 65 65
79 Eccentricity in the Sections eccentricity = 1
80 Eccentricity in the Sections The eccentricity of the parabola is 1.
81 Definition of the Parabola PF 1 /PQ = sin 65 /sin 65 P F 1 Q
82 Definition of the Parabola PF 1 = PQ P F 1 Q
83 Definition of the Parabola P F 1 Q For every point, distance to the focus equals distance to the directrix.
84 Eccentricity in the Sections 80 eccentricity = sin 80 /sin 65 65
85 Eccentricity in the Sections PF 1 /PQ = sin 80 /sin 65 F 1 P Q
86 Eccentricity in the Sections PF 1 /PQ = constant greater than 1 F 1 P Q
87 Geometry of the Steep Section Eccentricity of the hyperbola exceeds 1. F 1 P Q
88 Speed and Eccentricity 17,600 mph
89 Speed and Eccentricity circle eccentricity = (v/v 0 ) 2 1 = = 0 17,600 mph
90 Speed and Eccentricity 26,200 mi Add 32% 23,200 mph
91 Speed and Eccentricity ellipse eccentricity = (v/v 0 ) 2 1 = ,200 mi Add 32% 23,200 mph
92 Speed and Eccentricity ellipse eccentricity = (v/v 0 ) 2 1 = ,000 mi Add 39% 24,500 mph
93 Speed and Eccentricity eccentricity = (v/v 0 ) 2 1 = Add 41.4% 24,900 mph
94 Speed and Eccentricity parabola eccentricity = (v/v 0 ) 2 1 = ( 2) 2 1 = 1 Add 41.4% 24,900 mph
95 Speed and Eccentricity hyperbola eccentricity = (v/v 0 ) 2 1 = = 1.25 Add 50% 26,400 mph
96 Elements of the Parabola one axis F one directrix
97 Elements of the Parabola baseline F
98 Extent of the Parabola F no points below the baseline
99 Elements of the Parabola no points along the axis F no points below the baseline
100 Extent of the Parabola points in all other directions F
101 Elements of the Hyperbola F 1 one axis
102 Elements of the Hyperbola F 1 F 2 second focus and directrix
103 Elements of the Hyperbola F 1 second axis F 2
104 Elements of the Hyperbola F 1 F 2
105 Elements of the Hyperbola F 1 F 2
106 Extent of the Hyperbola Hyperbola is confined to the gray region. F 1 F 2
107 Reflection Properties: the Ellipse F 1 F 2
108 Reflection Properties: the Ellipse F 1 F 2
109 Reflection Properties: the Ellipse F 1 F 2
110 Reflection Properties: the Parabola P F
111 Reflection Properties: the Parabola P F
112 Reflection Properties: the Hyperbola F 1 F 2
113 Reflection Properties: the Hyperbola F 1 F 2
114 Reflection Properties: the Hyperbola F 1 F 2
115 Reflection Properties: the Hyperbola F 1 F 2
116 Telescopes and the Conics
117 Telescopes and the Conics
118 Telescopes and the Conics
119 Telescopes and the Conics
120 Telescopes and the Conics
121 Telescopes and the Conics
(3,4) focus. y=1 directrix
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