Abstract
From the analytic viewpoint of Klein’s definition of geometry, projective geometry is the logical generalization of the affine geometry introduced in Chapter 3. Just as we were able to generalize the isometries of the Euclidean plane to similarities, and these in turn to affinities, we will now be able to generalize affinities to collineations, the transformations that define projective geometry. There is, however, one new ingredient required in this last generalization. The set of points contained in the Euclidean plane must be enlarged to include points on one additional line, a line often referred to as the ideal line. Rather than complicating the geometry, these new ideal points simplify projective geometry and give it the highly desirable property of duality.
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Suggestions for Further Reading
Coxeter, H.S.M. (1957). Non-Euclidean Geometry, 3rd ed. Toronto: University of Toronto Press. (Includes a detailed presentation of Euclidean and non-Euclidean geometries as subgeometries of projective geometry. )
Coxeter, H.S.M. (1961). The Real Projective Plane, 2nd ed. Cambridge: The University Press. (A primarily synthetic presentation restricted to the real plane, it includes the development of affine geometry. )
Coxeter, H.S.M. (1987). Projective Geometry, 2nd ed. New York: Springer-Verlag. (A classic text containing a detailed development of this geometry. )
Dorwart, H. (1966). The Geometry of Incidence. Englewood Cliffs, NJ: Prentice-Hall. (An expository overview of projective geometry. )
Kline, M. (1968). Projective geometry. In Mathematics in the Modern World: Readings from Scientific American, pp. 120–127. San Francisco: W.H. Freeman. (A short, easyto-read introduction.)
Meserve, B.E. (1983). Fundamental Concepts of Geometry. New York: Dover. (Chap- ters 5 and 8 give a more detailed presentation of the material in Section 4.12.)
Mihalek, R.J. (1972). Projective Geometry and Algebraic structures. New York: Academic Press. (A detailed presentation emphasizing the interrelation between geometry and algebra. )
Pedoe, D. (1963). An Introduction to Projective Geometry. Oxford: Pergamon Press. (Contains an extensive treatment of the theorems of Desargues and Pappus.) Penna, M.A., and Patterson, R.R. (1986). Projective Geometry and Its Applications to Computer Graphics. Englewood Cliffs, NJ: Prentice-Hall.
Seidenberg, A. (1962). Lectures in Projective Geometry. New York: Van Nostrand Reinhold. (The initial chapter introduces the major concepts in a fairly naive form; the remaining chapters develop the subject from axioms. )
Stevenson, F.W. (1972). Projective Planes. San Francisco: W.H. Freeman.
Tuller, A. (1967). Modern Introduction to Geometries. New York: Van Nostrand Reinhold. (Uses matrix representations of the projective transformations. )
Wylie, C.R. Jr. (1970). Introduction to Projective Geometry. New York: McGraw-Hill. (Contains both analytic and axiomatic developments. )
Young, J.W. (1930). Projective Geometry. The Carus Mathematical Monographs, No. 4. Chicago: Open Court Publishing Co. (for the M.A.A.). (Develops concepts intuitively first and then incorporates metric properties and group concepts.)
Readings on the History of Projective Geometry
Bronowski, J. (1974). The music of the spheres. In: The Ascent of Man, pp. 155–187. Boston: Little, Brown. This chapter is the companion to the 52-minute episode of the same name in The Ascent of Man television series.
Edgerton, S.Y. (1975). The Renaissance Rediscovery of Linear Perspective. New York: Basic Books.
Ivins, W.M. (1964). Art and Geometry: A Study in Space Intuitions. New York: Dover.
Kline, M. (1963). Mathematics: A Cultural Approach. Reading, MA: Addison-Wesley.
Kline, M. (1968). Projective geometry. In: Mathematics in the Modern World: Readings from Scientific American, pp. 120–127. San Francisco: W.H. Freeman.
Kline, M. (1972). Mathematical Thought from Ancient To Modern Times. New York: Oxford University Press.
Pedoe, D. (1983). Geometry and the Visual Arts. New York: Dover.
Suggestions for Viewing
Central Perspectivities (1971, 13.5 min). Demonstrates perspectivities and projectivities with flashing dots and lines. Produced by the College Geometry Project at the University of Minnesota. Available from International Film Bureau, 332 South Michigan Avenue, Chicago, IL 60604.
Projective Generation of Conics (1971, 16 min). Illustrates four methods of constructing point conics and demonstrates their logical equivalence. Available from International Film Bureau, 332 South Michigan Avenue, Chicago, IL 60604.
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Cederberg, J.N. (1989). Projective Geometry. In: A Course in Modern Geometries. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3831-5_4
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DOI: https://doi.org/10.1007/978-1-4757-3831-5_4
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