Spherical
Geometry:

Exploring the World with Math

Exploring the World with Math

* Ptolemy*:
Picture courtesy of
Almagest Ephemeris
Calculator

**Karen Franco**

kffranco(at)interchange(dot)ubc(dot)ca

Student # 46347985

MATH 308, Section 102

Final Project

December 15, 2002

**Table of Contents:**

I.
Introduction

II.
The Basics of Spherical Geometry

III.
Great Circles

IV.
Spherical Triangles

A Tale of Two Cities: The Cross-Continental Application of the Solution of
Spherical Triangles

V.
Conclusion

References

**I. Introduction**

Captain Cook, mathematician? It is a little known fact that Captain James Cook,
discoverer of Australia, New Zealand, Papua New Guinea, Hawaii, Tahiti and other
islands in the Pacific, was trained as both a navigator and as a mathematician.
In fact, mathematics and exploration have a long history dating back to the
times of the Greek and Phoenician mariners.

In the modern world, mathematics is commonly regarded as a "sit-down" science --
a subject where problems are often solved while sitting down in a classroom or
office, and applications often relate to theory, finance, or business. However,
during the days of exploration, when it was discovered that the world was indeed
round and not flat, * spherical geometry* was integral in mapping out
the world, in navigating the seven seas, and in using the position of stars to
chart courses from one continent to another.

A sphere is defined as a closed surface in 3D formed by a set of points an equal distance

An arbitrary straight line (not lying in the sphere) and sphere in three dimensional space can either (a) not intersect at all; (b) intersect at one point on the sphere, when the line is

Figure 1: Line passing through the centre
of the sphere; points of intersection are ** antipodes** (PostScript
file)

**III. Great Circles
**

Like lines and spheres, an arbitrary straight plane and sphere in three dimensional space can have (a) no intersection; (b) one point of intersection, when the plane is

Figure 2: The meridians of longitude are examples of **
great circles** (animated PostScript)

* Great circles* are defined as those circles of intersection which share
the same radius

Figure 3: The parallels of latitude are examples of *
small circles* (animated PostScript)

Imagine a line from the North to the South Pole, passing through the centre of
the globe. The circles of intersection formed by the globe and a plane
perpendicular to this imaginary line make up the globe's lines or parallels of
latitude. Each of these circles of intersection, with the exception of the
Equator at which point the plane is at the midpoint of the pole-to-pole line,
are called * small circles* precisely because their radii measure
less than the Earth's radius

Navigators often used great circles to figure out the most efficient route to their destinations. It turns out that the shortest path between two points on a sphere is along a great circle path, that is, along an arc of a great circle. Have you ever wondered why a plane flying from Vancouver to the Philippines follows a route that takes you over Japan and Korea instead of flying a straight line over the Pacific Ocean? Or why a flight from New York to Europe has to travel over the Maritimes and almost reach Greenland instead of making a beeline over the Atlantic Ocean? The exact reason behind the logic of taking great circle paths to travel the world is explained and proved in the following section.

When the arcs of three great circles intersect on the surface of a sphere, the lines enclose an area known as a

Have you ever heard of a triangle whose angles sum up to greater than 180°? In the figure below, the two meridians of longitude are separated by an angle of 90° and both lines of longitude fall perpendicular to the Equator (the only great circle of latitude). Each angle in this particular spherical triangle equals 90°, and the sum of all three add up to 270°.

Figure 4: In this triangle, the sum of the three angles exceeds 180° (and equals 270°)

Spheres have positive curvature (the surface curves outwards from the centre),
hence the sum of the three angles of a triangle exceeds 180°. On a plane with
zero curvature, the sum of a triangles angles equals exactly 180°.

Like their angles, the length of the sides of a spherical triangle are measured
in degrees or radians. Specifically, the length of a side of a spherical
triangle equals the measurement of its opposite angle. In geography, the angle
between two meridians of longitude equals the same number of degrees as the arc
cut off by these lines of longitude on any circle of latitude. So, in the above
figure, each of the sides measures 90° since each of their opposing angles
measures 90°.

The most useful application of spherical triangles and great circles is perhaps
calculating the shortest-distance route between two points on the globe. This
application is often referred to as the * solution of spherical triangles*
and makes extensive use of the well known Cosine Law for triangles on a plane:

Figure 5: The solution of a
sphere

- The spherical triangle
*abc*is formed by the intersections of great circles with planes intersecting at OA, OBQ and OCP. - The plane PQA is partly
composed of two tangent lines: AQ tangent to
*c*and AP tangent to*b*, and will be referred to as our tangent plane. - Therefore, OAQ and OAP
are right angles and PAQ equals the angle A opposite side
*a*.

Figure 6: The net of the tetrahedron used for the solution of a spherical triangle (animated PostScript)

- Extracting the
tetrahedron enclosed by the planes and laying it flat on a plane as a net, we
examine the 4 composite triangles:
- Triangles OAQ and OAP
are right-angled triangles, so using Pythagoras' Theorem:
- PO
^{2}= AO^{2}+ PA^{2}- AO
^{2}= PO^{2}- PA^{2}

- AO
- QO
^{2}= AO^{2}+ QA^{2}- AO
^{2}= QO^{2}- QA^{2}

- AO

- PO

- The other two
triangles, QAP and QOP are generic plane triangles, so using the Cosine Law
for flat triangles, we can see that
- PQ
^{2}= PO^{2}+ QO^{2}- 2 PO·QO cos*a* - PQ
^{2}= PA^{2}+ QA^{2}- 2 PA·QA cos A

- PQ
- Subtracting the two
equations above from each other, we get:
- (PO
^{2}- PA^{2}) + (QO^{2}- QA^{2}) - (2 PO·QO cos*a*- 2 PA·QA cos A) = (PQ^{2}- PQ^{2}) - (PO
^{2}- PA^{2}) + (QO^{2}- QA^{2}) - 2 PO·QO cos*a*+ 2 PA·QA cos A = 0

- (PO
- Substituting AO2 for
(PO2 - PA2) and (QO2 - QA2):
- 2 AO
^{2}+ 2 PA·QA cos A = 2 PO·QO cos*a*

- 2 AO
- Dividing through both
sides by 2 PO·QO:
- cos
*a*= (AO/PO)·(AO/QO) + (PA/PO)·(QA/QO) cos A

- cos
- But we know that
(AO/PO) = cos POA, (AO/QO) = cos QOA, (PA/PO) = sin POA, and (QA/QO) = sin
QOA
- cos
*a*= cos POA · cos QOA + sin POA · sin QOA · cos A

- cos
- Finally, substituting
the side opposite the spherical angle,
*b*for angle POA and*c*for angle QOA:- cos
*a*= cos*b*· cos*c*+ sin*b*· sin*c*· cos A

- cos

- Triangles OAQ and OAP
are right-angled triangles, so using Pythagoras' Theorem:

Therefore the formula for
the third side, *a*, of a spherical triangle, given two sides, *b* and
*c*, and their included angle, A is

cosa= cosb· cosc+ sinb· sinc· cos A

__A Tale of Two Cities: The Cross-Continental Application of the Solution of
Spherical Triangles__

Figure 7: Pictures courtesy of Maps.com

Imagine you had to find the best route from New York to London. New York is
geographically located at along the great circle of longitude 74° 0' W and
approximately 40° 42' latitude north of the Equator, which makes it 90° - 40°
42' = 49° 18' south of the North Pole. London, on the other hand, is located
along the great circle of longitude 0° 5' W at approximately 51° 32' north of
the Equator, which makes it 90° - 51° 32' = 38° 28' south of the North Pole.
The sides *b *and *c* are given by the length of the arcs from the
North Pole to New York and London respectively, so *b* = 49° 18' and *c*
= 38° 28'. The angle A is given by the difference in meridians of longitude for
the two cities: A = 74° 0' W - 0° 5' W = 73° 55'.

Applying the solution**
cos a = cos b · cos c + sin b · sin c · cos A**,
we get the following calculation:

cos

a= cos 49° 18' · cos 38° 28' + sin 49° 18' · sin 38° 28' · cos 73° 55'

cosa= (0.6521 · 0.7830) + (0.7581 · 0.6221 · 0.2770)

cosa= 0.6412

a= 50.1186° or 50° 7'

This means that the great
circle distance between New York and London is approximately 50° 7'. In miles,
given that one degree of a great circle is approximately 69 miles (110.4
kilometres), this distance measures approximately 50.1151° x 69 miles = 3458
miles (5533.0934 kilometres).

**V. Conclusion**

Geometry derives its meaning from the Greek words *geometria* and *
geometrein *which mean "measuring the earth". Geography, on the other hand
derives its meaning from the Greek words *geographia* and *geographein*
which mean "describing or writing about the earth". One would expect words so
similar in meaning to be similar in concept as well. However, the two fields
were separate and distinct until the days of ancient Greece, when Ptolemy
(astronomer, mathematician and geographer) made use of geometry in reasoning
more about the earth and its shape:

"In Geography one must contemplate the extent of the entire earth, as well as its shape, and its position under the heavens, in order that one may rightly state what are the peculiarities and proportions of the part with which one is dealing... It is the great and exquisite accomplishment of mathematics to show all these things to human intelligence..."

Interestingly enough, it was
also Ptolemy and not Christopher Columbus who discovered that the earth was
spherical and not flat, and stated his rationale in the *Almagest* 1300
years before Columbus sailed around the world:

"If the earth were flat from east to west, the stars would rise as soon for westerners as for orientals, which is false. Also, if the earth were flat from north to south and vice versa, the stars which were always visible to anyone would continue to be so wherever he went, which is false. But it seems flat to human sight because it is so extensive."

Like geometry and geography, the worlds of spherical geometry (used in
geography) and planar geometry (commonly taught in most geometry courses) are
closely related and yet extremely different.

Anybody who has completed high school level geometry (or to some extent,
elementary geometry) knows that in Euclidean or planar geometry, two parallel
lines never meet, the sum of the three angles of a triangle add up to 180°, and
the shortest route to get from one point to another is a straight line. In the
world of spherical geometry, two parallel lines on great circles intersect
twice, the sum of the three angles of a triangle on the sphere's surface exceed
180° due to positive curvature, and the shortest route to get from one point to
another is not a straight line on a map but a line that follows the minor arc of
a great circle. Maps provide a way of translating the spherical view of the
world to a planar view, by projecting the Earth's topologies and locations to a
flattened surface using Hammer, Mercator or cylindrical methods. A consistent
and standard representation that minimizes projective distortions is yet to be
established.

The discovery of spherical geometry not only changed the history and the face of
mathematics and Euclid's geometry, but also changed the way humans viewed and
charted the world. Using this new knowledge, explorers and astronomers used the
circular path of stars to navigate the earth to discover new lands and reason
about the cosmos.

**
References:
**Borowski, E.J. and Borwein,
J.M.

Casselman, Dr. W.

Hogben, Lancelot.

Hogben, Lancelot.

Maps.com - Learn and Play.
__Map Skills: Great Circles.__ [Maps.com
web page]

Mariners' Museum, The. __The Mariners' Museum -- Newport News, Virginia.__
[background image for web
page, historical background]

Osserman, Robert. __Poetry of the Universe: A Mathematical Exploration of the
Cosmos.__ 1995: Anchor Books, Doubleday. New York.

Polking, John C. __The Geometry of the Sphere 1.__ [basic
information about spheres]

WhatIs?Com. __Latitude and Longitude: a WhatIs Definition.__ [what
is latitude and longitude?]

Wolfram Research, Inc. __MathWorld: Eric Weisstein's World of Mathematics.__
[MathWorld web
page]

**TROUBLESHOOTING:**

**In order to run any of
the PostScript files, you need a free PostScript interpreter installed on your
computer: GhostView and
GhostScript are the most popular
ones. Click on the links to obtain these.**

**For latitude.ps and
longitude.ps, make sure you have ps3d.inc in the same directory as these files
(see KarenFrancoProject.zip file if this is missing).**