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In a historical perspective on mathematics, the field of geometry that developed in the first half of the nineteenth century under the name
projective geometry was a stepping stone from analytic geometry to algebraic
geometry. When treated in terms of homogeneous
co-ordinates it looks like an extension or technical improvement of the use of co-ordinates to reduce geometric problems to
algebra, that reduced the number of special cases. And on the other hand the detailed
study of quadrics and the 'line geometry' of Julius Plucker still forms a rich set of examples for geometers who also work
with more general concepts. Towards the end of the century the Italian school (Enriques, Segre, Severi) broke out of the traditional subject matter into an area demanding deeper techniques.
It should be said first that the notable projective geometers, including Poncelet, Steiner and others, were not
intending to extend analytic geometry. Techniques were supposed
to be synthetic: in effect projective space as we understand it now was to be introduced on an
axiomatic basis. This poses some problems in recovering the theory. In the case of the projective plane alone, the axiomatic approach may encounter models that cannot be described via linear algebra.
Whatever the 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 in exactly one point P. The
special case in analytic geometry of parallel lines has
been subsumed in the smoother form of a line at infinity on which P will lie in that case. 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).
It also included a full theory of conic sections, a subject that
already had a large number of theorems (mainly useful as a source of examination questions). There are great and 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 allowing complex number co-ordinates. Since co-ordinates were not 'synthetic', one replaces that 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 whole approach was very attractive to talented geometers, and the field was thoroughly worked over. A later
many-volume work by H.F. Baker shows the style.
This period in geometry was rather overtaken by the research on the general algebraic curve by Clebsch, Riemann, Max Noether and others, which stretched
existing techniques, and then by invariant theory. In the latter
part of the nineteenth century, the detailed study of projective geometry itself was less important to professional
mathematicians, though the literature is voluminous. Some important work was done in enumerative geometry in
particular, by Schubert, that is now considered an anticipation of the theory of Chern classes in their guise as representing the algebraic topology of Grassmannians.
See also
Projective plane, projective space, projective
transformation, homogeneous coordinates, Desargues' theorem, Möbius transformation, incidence, cross-ratio, duality.
References
- Coxeter, H. S. M., The Real Projective Plane, 3rd ed,
Springer Verlag, New York, 1995
- Coxeter, H. S. M., Projective Geometry, 2nd ed., Springer Verlag, New York, 2003
- Veblen, O. and Young, J. W., Projective Geometry, 2 vols., Blaisdell Pub. Co., New York, 1938-46
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