Where is spherical geometry used

In spherical geometry, because there are no parallel lines, these two perpendiculars must intersect. But there is something more subtle involved in this third postulate.

All perpendiculars meet at the same point. In this geometry, several counter-intuitive results follow. Some of these results are the same as the results in hyperbolic geometry, but for different reasons. All rights reserved including the right of reproduction in whole or in part in any form.

To order this book direct from the publisher, visit the Penguin USA website or call You can also purchase this book at Amazon. Figure Tangent Line Spherical geometry is important in navigation, because the shortest distance between two points on a sphere is the path along a great circle.

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We reserve the right to remove any content at any time from this Community, including without limitation if it violates the Community Standards. Given two sides of a spherical triangle and the angle between these sides, the solution for a spherical triangle yields the length of the third side. Figure 5: The solution of a sphere. Figure 6: The net of the tetrahedron used for the solution of a spherical triangle animated PostScript.

Therefore the formula for the third side, a , of a spherical triangle, given two sides, b and c , and their included angle, A is. Figure 7: Pictures courtesy of Maps. Imagine you had to find the best route from New York to London.

In miles, given that one degree of a great circle is approximately 69 miles 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:. 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 years before Columbus sailed around the world:.

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.

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. Collins Reference Dictionary of Mathematics. London and Glasgow. Casselman, Dr. A Manual of Mathematical Illustration. Mathematics for the Million.