Abstract
In this chapter we discuss the non-Euclidean geometry of curved surfaces, using the sphere as our primary example. We find that all the information about the geometry of the surface is contained in the expression for the distance between two nearby points in some coordinate system, called the metric. For example, the distance between two distant points can be found from the metric by determining and measuring the minimum length curve (geodesic) which connects them.
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Notes
- 1.
Gerardus Mercator (1512–1594), Flemish cartographer.
- 2.
Nikolai Ivanovich Lobachevsky (1792–1856), Russian mathematician.
- 3.
János Bolyai (1802–1860), Hungarian mathematician.
- 4.
Carl Friedrich Gauss (1777–1855), German mathematician and astronomer.
- 5.
Georg Bernhard Riemann (1826–1866), German mathematician.
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© 2011 Springer-Verlag Berlin Heidelberg
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Natário, J. (2011). Non-Euclidean Geometry. In: General Relativity Without Calculus. Undergraduate Lecture Notes in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21452-3_3
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DOI: https://doi.org/10.1007/978-3-642-21452-3_3
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21451-6
Online ISBN: 978-3-642-21452-3
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