Projective geometry

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Projective geometry is a non-metrical form of geometry that emerged in the early 19th century.

1 Description
2 History
3 Forms of the living world
4 See also
5 References
6 External sites

Description

Projective geometry can be formulated as an axiomatic first order theory (with identity), whose universe contains "points" and "lines." Hence there are two primitive sets, one whose members are the points and the other whose members are the lines. There is a primitive binary relation, called "incidence," linking points and lines, denoted by the preposition "on": point P is on line L. Objects A and B are "distinct" if A=B is false. The axioms are (Eves 1997: 111):

Any two distinct points are on exactly one line;
There are at least three distinct points on any line;
Given any line, there exists a point not on the line;
Given any two distinct lines, there exists a point that is on both of them (any two distinct lines have a common point).

All infinite geometries can be derived as special cases of projective geometry, by adding the requisite primitive notions and axioms.

In 1825, Joseph Gergonne noted the principle of duality characterizing projective plane geometry: given any theorem or definition of that geometry, substituting point for line, lie on for pass through, collinear for concurrent, intersection for join, or vice versa, results in another theorem or valid definition, the "dual" of the first. In the projective geometry of three dimensional space, duality holds for points and planes. Hence all theorems and definitions in projective geometry come in "dual pairs." To establish duality, verify that the axioms imply their own duals. It is useful to verify that the dual of an axiom is either an axiom or is derivable from the remaining axioms.

Informally, projective geometry arises from placing one's eye at a point (or perspective). That is, every line which intersects the "eye" appears only as a point in the projective plane because the eye cannot "see" the points behind it. Projective geometry is also a useful way of proving certain theorems of Euclidean geometry.

Whatever its precise foundational status, projective geometry did include basic incidence properties. That means that any two distinct lines L and M in the projective plane intersect at exactly one point P. The special case in analytic geometry of parallel lines is subsumed in the smoother form of a line at infinity on which P lies. The point is then that the line at infinity is a line like any other in the theory: it is in no way special or distinguished. (In the later spirit of the Erlangen programme one could point to the way the group of transformations can move any line to the line at infinity).

Projective geometry also includes a full theory of conic sections, a subject already very well developed in Euclidian geometry (and mainly useful as a source of examination questions). There are clear advantages in being able to think of a hyperbola and an ellipse as distinguished only by the way the hyperbola lies across the line at infinity; and that a parabola is tangent to the same line. The whole family of circles can be seen as the conics passing through two given points on the line at infinity - at the cost of requiring complex number coordinates. Since coordinates were not "synthetic", one replaces them by fixing a line and two points on it, and considering the linear system of all conics passing through those points as the basic object of study. This approach proved very attractive to talented geometers, and the field was thoroughly worked over. An example of this approach is the multi-volume treatise by H. F. Baker.

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History

Projective geometry originated through the efforts of a French artist and mathematician, Gerard Desargues (1591-1666), as an alternative way of constructing perspective drawings. By generalizing the use of vanishing points to include the case when these are infinitely far away, he made Euclidean geometry, where parallel lines are truly parallel, into a special case of an all-encompassing geometric system. The work of Desargues was totally ignored until Michel Chasles chanced upon a handwritten copy in 1845. Meanwhile, Jean-Victor Poncelet had published the foundational treatise on projective geometry in 1822. The non-Euclidean geometry discovered shortly thereafter was eventually demonstrated to be a special case of projective geometry.

This early 19th century projective geometry was a stepping stone from analytic geometry to algebraic geometry. When treated in terms of homogeneous coordinates, projective geometry looks like an extension or technical improvement of the use of coordinates to reduce geometric problems to algebra, an extension reducing the number of special cases. The detailed study of quadrics and the 'line geometry' of Julius Plücker still form a rich set of examples for geometers working with more general concepts.

The work of Poncelet, Steiner and others was not intended to extend analytic geometry. Techniques were supposed to be synthetic: in effect projective space as now understood was to be introduced axiomatically. As a result, reformulating early work in projective geometry so that it satisfies current standards of rigor can be somewhat difficult. Even in the case of the projective plane alone, the axiomatic approach can result in models not describable via linear algebra.

This period in geometry was overtaken by research on the general algebraic curve by Clebsch, Riemann, Max Noether and others, which stretched existing techniques, and then by invariant theory. Towards the end of the century the Italian school of algebraic geometry (Enriques, Segre, Severi) broke out of the traditional subject matter into an area demanding deeper techniques.

In the latter part of the 19th century, the detailed study of projective geometry became less important, although the literature is voluminous. Some important work was done in enumerative geometry in particular, by Schubert, that is now seen as anticipating the theory of Chern classes, taken as representing the algebraic topology of Grassmannians.

Hermann von Baravalle has explored the pedagogical potential of projective geometry for school mathematics.

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Forms of the living world

In the spirit of projective geometry's origins in synthetic geometry, some mathematicians have investigated projective geometry as a useful way of describing natural phenomena. The first research in this direction was stimulated by a suggestion by the philosopher Rudolf Steiner (not to be confused with the mathematician Jakob Steiner, mentioned above).

In the mid-twentieth century,, Louis Locher-Ernst explored the tension between central forces and peripheral influences. Lawrence Edwards (*1912-d.2004) discovered significant applications of Klein path curves to organic development. In the spirit of D'Arcy Thompson's On Growth and Form, but with more mathematical rigor, Edwards demonstrated that such forms as the buds of leaves and flowers, pine cones, eggs, and the human heart can be simply described by certain path curves. Varying a single parameter, lambda, metamorphoses the interaction of what are known in projective geometry as growth measures into surprisingly accurate representations of many organic forms not otherwise easily describable mathematically; negative values of the same parameter produce inversions representing vortexes of both water and of air.

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References

Coxeter, H. S. M., 1995. The Real Projective Plane, 3rd ed. Springer Verlag.
--------, 2003. Projective Geometry, 2nd ed., Springer Verlag.
Hartshorne, Robin, 2000. Geometry: Euclid and Beyond. Springer.
Edward, Lawrence, Projective Geometry.
--------, The Vortex of Life.
Howard Eves, 1997. Foundations and Fundamental Concepts of Mathematics, 3rd ed. Dover.
Locher-Ernst, Louis, Space and Counterspace.
Oswald Veblen and J. W. A. Young, 1938-46. Projective Geometry, 2 vols. New York: Blaisdell.