Abstract
Modern geometry developed from the work of Euclid which gave rise to two self-contained geometries, known as absolute geometry and affine geometry. Euclidean geometry depends on five postulates:
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1.
A straight line may be drawn from any point to any other point.
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2.
A finite straight line may be extended continuously in a straight line.
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3.
A circle may be described with any centre and any radius.
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4.
All right angles are equal to each other.
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5.
If a straight line meets two other straight lines so as to make the two interior angles on one side of it together less than two right angles, the other straight lines, if extended indefinitely, will meet on that side on which angles are less than two right angles.
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Notes
- 1.
It will in fact be shown that all conics, including the straight line, are equivalent in projective geometry.
- 2.
It will be shown that physical models formulated in five-dimensional Euclidean (or affine) space are reproduced in four-dimensional projective space without the awkward concept of compacted dimensions.
- 3.
Notice how a flat sheet of paper can be rolled into a cylinder or a cone (K=0) without getting crumpled.
- 4.
Although it is customary to refer to covariant and contravariant vectors, this may be misleading. Any vector can be described in terms of its contravariant or its covariant components with equal validity. There is no reason other than numerical simplicity for the preference of one set of components over the other.
References
Coxeter, H.S.M. (1998): Introduction to Geometry, 2nd ed., Wiley, N.Y.
Flegg, H.G. (1974): From Geometry to Topology, Reprinted 2001, Dover, Mineola, NY.
Jennings, G.A. (1994): Modern Geometry with Applications, Springer, N.Y.
Lee, J.M. (1997): Riemannian Manifolds, Springer, N.Y.
Veblen, O. & J.W. Young (1910): Projective Geometry, Vol.I, 1910; Vol.II (O. Veblen sole author), 1918, Ginn and Co., Boston.
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Boeyens, J.C.A. (2010). World Geometry. In: Chemical Cosmology. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3828-9_3
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DOI: https://doi.org/10.1007/978-90-481-3828-9_3
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