Abstract
Physical space can be represented by a formal geometric space, which can be further represented by a numeric coordinate space. Euclidean geometry is constructed from point elements of no size, with real-number coordinates, and has unlimited extent. Discrete geometry is constructed from cube elements of fixed size, with integer coordinates, and unlimited extent. Harmonic space has a harmonic scale of distances, and lies within a finite octahedron. Elliptic space has homogeneous coordinates, and lies within a finite sphere.
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Notes
- 1.
Felix Christian Klein (1849–1925), Professor at Erlangen, Munich, Leipzig, and Göttingen.
- 2.
Renée Descartes (1596–1650): born in France; lived in the Dutch Republic 1628–1649; taught at Utrecht University and elsewhere. The same idea was used, possibly as early as 1628, by Pierre de Fermat (1607–1665), lawyer in Toulouse and distinguished mathematician.
- 3.
Almost nothing is known about Euclid the man, beyond the possibility that he studied under Plato in Athens, and the certainty that he taught in Alexandria around 300Â BC.
- 4.
Karl Georg Christian von Staudt (1798–1867), Professor of Mathematics at Erlangen University. The construction was invented to assign rational coordinates to points in a projective plane, using incidence alone. The result, which can never be completed in practice, is often called a net of rationality, or a Mobius net.
- 5.
The ancient Greeks were very uncomfortable with the unlimited (\(\alpha \pi \varepsilon \iota \rho \omega \nu \)), as they were with the irrational (\(\surd 2\)). The modern conventions for inclusion of \(\infty \) in arithmetic are: \(n + \infty = \infty \); \(n - \infty = -\infty \); \(n \times \infty = \infty \); \(n / \infty = 0\); \(\infty /n = \infty \); \(n/0 = \infty \).
- 6.
Introduced in 1827 by August Ferdinand Mobius (1790–1868), Professor at Leipzig University.
References
Descartes R (1637) La Géométrie, Appendix to Discourse de la Méthode. Jan Maire, Leyden. http://www.gutenberg.org/etext/26400. Accessed 1 Feb 2014
Heath TL (1955) The thirteen books of Euclid’s elements (trans: heath TL), vol 3. Dover, New York
Klein FC (1872) Review of recent researches in geometry (trans: Haskell MW 1892-3). http://arxiv.org/pdf/0807.3161v1.pdf. Accessed 1 Feb 2014
Staudt KGC von (1847) Geometrie der Lage, Bauer und Raspe, Nürnberg. Reprinted 2011 by Nabu Press
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Parkin, A. (2016). Coordinate Geometry. In: Digital Imaging Primer. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85619-1_6
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DOI: https://doi.org/10.1007/978-3-540-85619-1_6
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