Abstract
Geometrically, curves and surfaces are usually considered to be sets of points with some special properties, living in a space consisting of “points.” Typically, one is also interested in geometric properties invariant under certain transformations, for example, translations, rotations, projections, etc. One could model the space of points as a vector space, but this is not very satisfactory for a number of reasons. One reason is that the point corresponding to the zero vector (0), called the origin, plays a special role, when there is really no reason to have a privileged origin. Another reason is that certain notions, such as parallelism, are handled in an awkward manner. But the deeper reason is that vector spaces and affine spaces really have different geometries. The geometric properties of a vector space are invariant under the group of bijective linear maps, whereas the geometric properties of an affine space are invariant under the group of bijective affine maps, and these two groups are not isomorphic. Roughly speaking, there are more affine maps than linear maps.
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References
Michael Artin. Algebra. Prentice-Hall, first edition, 1991.
Marcel Berger. G´eom´etrie 1. Nathan, 1990. English edition: Geometry 1, Universitext, Springer-Verlag.
Marcel Berger. G´eom´etrie 2. Nathan, 1990. English edition: Geometry 2, Universitext, Springer-Verlag.
H.S.M. Coxeter. Introduction to Geometry. Wiley, second edition, 1989.
Jean H. Gallier. Curves and Surfaces in Geometric Modeling: Theory and Algorithms. Morgan Kaufmann, first edition, 1999.
Donald T. Greenwood. Principles of Dynamics. Prentice-Hall, second edition, 1988.
D. Hilbert and S. Cohn-Vossen. Geometry and the Imagination. Chelsea Publishing Co., 1952.
Serge Lang. Algebra. Addison-Wesley, third edition, 1993.
Dan Pedoe. Geometry, A Comprehensive Course. Dover, first edition, 1988.
Pierre Samuel. Projective Geometry. Undergraduate Texts in Mathematics. Springer-Verlag, first edition, 1988.
Ernst Snapper and Troyer Robert J. Metric Affine Geometry. Dover, first edition, 1989.
Gilbert Strang. Linear Algebra and Its Applications. Saunders HBJ, third edition, 1988.
Claude Tisseron. G´eom´etries Affines, Projectives, et Euclidiennes. Hermann, first edition, 1994.
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Gallier, J. (2011). Basics of Affine Geometry. In: Geometric Methods and Applications. Texts in Applied Mathematics, vol 38. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9961-0_2
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DOI: https://doi.org/10.1007/978-1-4419-9961-0_2
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