Spherical Geometry and World Navigation

1. Spherical Geometry and World Navigation By Houston Schuerger

2. 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

3. Euclid’s Five Axioms • 1. A straight line segment can be drawn joining any two points.

4. 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.

5. 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.

6. 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.

7. 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. • 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.

8. Euclid’s 5th 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

9. Euclid’s 5th 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.

10. Euclid’s 5th 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.”

11. Euclid’s 5th 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.

12. 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

13. 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.

14. 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

15. 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

16. 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.

17. 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.

18. 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.

19. 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.

20. 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.

21. 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.

22. 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.