A Course in Modern Geometries pp 213-313 | Cite as

# Projective Geometry

## 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.

## Keywords

Projective Plane Ideal Point Projective Geometry Invariant Point Point Conic## Preview

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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.)MATHGoogle Scholar - 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.)Google Scholar - Coxeter, H.S.M. (1987).
*Projective Geometry*, 2nd ed. New York: Springer-Verlag. (A classic text containing a detailed development of this geometry.)MATHGoogle Scholar - Dorwart, H. (1966).
*The Geometry of Incidence*. Englewood Cliffs, NJ: Prentice-Hall. (An expository overview of projective geometry.)MATHGoogle Scholar - Meserve, B.E. (1983).
*Fundamental Concepts of Geometry*. New York: Dover. (Chapters 5 and 8 give a more detailed presentation of the material in Section 4.12.)Google Scholar - Mihalek, R.J. (1972).
*Projective Geometry and Algebraic Structures*. New York: Academic Press. (A detailed presentation emphasizing the interrelation between geometry and algebra.)MATHGoogle Scholar - Pedoe, D. (1963).
*An Introduction to Projective Geometry*. Oxford: Pergamon Press. (Contains an extensive treatment of the theorems of Desargues and Pappus.)MATHGoogle Scholar - Penna, M.A., and Patterson, R.R. (1986).
*Projective Geometry and Its Applications to Computer Graphics*. Englewood Cliffs, NJ: Prentice-Hall.MATHGoogle Scholar - Seidenberg, A. (1962).
*Lectures in Projective Geometry*. New York: Van Nos-trand Reinhold. (The initial chapter introduces the major concepts in a fairly naive form: the remaining chapters develop the subject from axioms.)MATHGoogle Scholar - Stevenson, E.W. (1972).
*Projective Planes*. San Francisco: W.H. Freeman.MATHGoogle Scholar - Tuller, A. (1967).
*Modern Introduction to Geometries*. New York: Van Nostrand Reinhold. (Uses matrix representations of the projective transformations.)MATHGoogle Scholar - Wylie, C.R. Jr. (1970).
*Introduction to Projective Geometry*. New York: McGraw-Hill. (Contains both analytic and axiomatic developments.)MATHGoogle Scholar - Young, J.W. (1930).
*Projective Geometry*. The Carus Mathematical Monographs, No. 4. Chicago: Open Court Publishing Co. (for the MAA) (Develops concepts intuitively first, then incorporates metric properties and group concepts.)MATHGoogle Scholar

## Readings on the History of Projective Geometry

- Bronowski, J. (1974). The music of the spheres. In
*The Ascent of Man, pip*. 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.Google Scholar - Edgerton, S.Y. (1975).
*The Renaissance Rediscovery of Linear Perspective*. New York: Basic Books.Google Scholar - Ivins, W.M. (1964).
*Art and Geometry: A Study in Space Intuitions*. New York: Dover.Google Scholar - Kline, M. (1963).
*Mathematics: A Cultural Approach*. Reading, MA: Addison-Wesley.Google Scholar - Kline, M. (1968). Projective geometry.
*In Mathematics in the Modern World: Readings from Scientific American*, pp. 120–127. San Francisco: W.H. Freeman.Google Scholar - Kline, M. (1972).
*Mathematical Thought from Ancient to Modern Times*. New York: Oxford University Press.MATHGoogle Scholar - Pedoe, D. (1983).
*Geometry and the Visual Arts*. New York: Dover.Google Scholar

## Suggestions for Viewing

*The Art of Renaissance Science*(1991, 45 min). Part IV relates the discovery and implementation of perspective in drawing and painting. Available from Science TV, P.O. Box 2498, Times Square Station, New York, NY 10108.Google Scholar*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.Google Scholar*Masters of Illusion*(1991, 30 min). Another illustration of the discovery and implementation of perspective in works of art. Produced and directed by Rick Harper, National Gallery of Art, Washington, DC.Google Scholar*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.Google Scholar*Projective Geometry*, Zeeman Masterclass Series with BBC (1986). Available from The Open University Production Centre, Walton Hall, Milton Keynes MK7 6BH, UK.Google Scholar