Transcript Spherical Geometry and World Navigation

By Houston Schuerger
Euclidean Geometry
 Most people are familiar with it
 Children learn shapes: triangles, circles, squares, etc.
 High school geometry: theorems concerning parallelism,
congruence, similarity, etc.
 Common, easy to understand, and abundant with
applications; but only a small portion of geometry
Euclid’s Five Axioms
 1. A straight line segment can be drawn joining any two
points.
Euclid’s Five Axioms
 1. A straight line segment can be drawn joining any two
points.
 2. Any straight line segment can be extended indefinitely
in a straight line.
Euclid’s Five Axioms
 1. A straight line segment can be drawn joining any two
points.
 2. Any straight line segment can be extended indefinitely
in a straight line.
 3. Given any straight line segment, a circle can be drawn
having the segment as radius and one endpoint as center.
Euclid’s Five Axioms
 1. A straight line segment can be drawn joining any two
points.
 2. Any straight line segment can be extended indefinitely
in a straight line.
 3. Given any straight line segment, a circle can be drawn
having the segment as radius and one endpoint as center.
 4. All right angles are congruent.
Euclid’s Five Axioms
 1. A straight line segment can be drawn joining any two
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points.
2. Any straight line segment can be extended indefinitely
in a straight line.
3. Given any straight line segment, a circle can be drawn
having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a
way that the sum of the inner angles on one side is less
than two right angles, then the two lines inevitably must
intersect each other on that side if extended far enough.
Euclid’s
th
5
Axiom
 more common statement equivalent to Euclid’s 5th axiom
 given any straight line and a point not on it, there exists one
and only one straight line which passes through that point
parallel to the original line
Euclid’s
th
5
Axiom
 more common statement equivalent to Euclid’s 5th axiom
 given any straight line and a point not on it, there exists one
and only one straight line which passes through that point
parallel to the original line
 5th axiom has always been very controversial
 Altering this final axiom yields non-Euclidean geometries, one
of which is spherical geometry.
Euclid’s
th
5
Axiom
 more common statement equivalent to Euclid’s 5th axiom
 given any straight line and a point not on it, there exists one
and only one straight line which passes through that point
parallel to the original line
 5th axiom has always been very controversial
 Altering this final axiom yields non-Euclidean geometries, one
of which is spherical geometry.
 This non-Euclidean geometry was first described by Menelaus
of Alexandria (70-130 AD) in his work “Sphaerica.”
Euclid’s
th
5
Axiom
 more common statement equivalent to Euclid’s 5th axiom
 given any straight line and a point not on it, there exists one




and only one straight line which passes through that point
parallel to the original line
5th axiom has always been very controversial
Altering this final axiom yields non-Euclidean geometries, one
of which is spherical geometry.
This non-Euclidean geometry was first described by Menelaus
of Alexandria (70-130 AD) in his work “Sphaerica.”
Spherical Geometry’s 5th Axiom: Given any straight line
through any point in the plane, there exist no lines parallel to
the original line.
Great Circles
 Straight lines of spherical
geometry
 circle drawn through the
sphere that has the same radii
as the sphere
 Occurs when a plane
intersects a sphere through its
center
 Shortest distance between
two points is along their
shared great circle
Spherical Geometry and World
Navigation
 The fact that great circles are the straight lines of spherical
geometry has a very interesting effect on world
navigation.
Spherical Geometry and World
Navigation
 The fact that great circles are the straight lines of spherical
geometry has a very interesting effect on world
navigation.
 Earth is not a perfect sphere, but it is much more similar to
a sphere than to the flat planes discussed in Euclidean
geometry
 Spherical geometry is far more appropriate to use when
discussing world navigation
Spherical Geometry and World
Navigation
 The fact that great circles are the straight lines of spherical
geometry has a very interesting effect on world
navigation.
 Earth is not a perfect sphere, but it is much more similar to
a sphere than to the flat planes discussed in Euclidean
geometry
 Spherical geometry is far more appropriate to use when
discussing world navigation
 Since great circles are the straight lines of spherical
geometry the shortest distance between two points is
along a great circle path
Spherical Geometry and World
Navigation
 When traveling a short distance the difference between
what appears to be a straight line connecting two points on
a map of the world and the great circle connecting the two
points is small enough that it can be ignored.
Spherical Geometry and World
Navigation
 When traveling a short distance the difference between
what appears to be a straight line connecting two points on
a map of the world and the great circle connecting the two
points is small enough that it can be ignored.
 When traveling a long distance such as the distance
between two continents the difference can be quite
substantial and costly to the uneducated navigator.
Spherical Geometry and World
Navigation
 If two cities on a globe lie on
the same latitudinal line it might
seem intuitive that travel
between the two cities would be
done along said latitudinal line.
Spherical Geometry and World
Navigation
 If two cities on a globe lie on
the same latitudinal line it might
seem intuitive that travel
between the two cities would be
done along said latitudinal line.
 However unless the latitudinal
line in question is the equator
then there will always be a
shorter path.
Spherical Geometry and World
Navigation
 If two cities on a globe lie on
the same latitudinal line it might
seem intuitive that travel
between the two cities would be
done along said latitudinal line.
 However unless the latitudinal
line in question is the equator
then there will always be a
shorter path.
 This is because even though all
longitudinal lines are great
circles the only latitudinal line
that is a great circle is the
equator.
Spherical Geometry and World
Navigation
 It is often the case that
these Great Circle paths
seem odd especially as one
tries to connect cities that
are far apart and far north
or south of the equator.
This is because the great
circle paths that connect
northern cities tend to
“curve” towards the North
Pole and southern cities
have a similar occurrence.
Spherical Geometry and World
Navigation
 For instance even though Tokyo
and St. Louis are both very
close to being located on the
37th parallel (St. Louis, 38° 40’
North 90° 15’ West; Tokyo 35°
39’ North 139° 44’ East) the
great circle which connects
them passes over Nome, Alaska
which is near the 64th parallel.
 Even though this still surprises
most people great circle routes
and their application to
navigation were first described
by Ptolemy in his work
Geographia in the year 150 AD.