In
mathematics projective geometry is the study of geometric properties which are invariant under
projective transformations. The field of projective geometry is itself divided into many subfields, two examples of which are projective algebraic geometry (the study of
projective varieties) and projective differential geometry (the study of
differential invariants of the projective transformations).
Projective geometry, like affine and Euclidean geometry, can be developed from the Erlangen program of Felix Klein. As such its geometric properties are invariant under the group action of the group of projective transformations. In Klein's Erlangen program, projective geometry is characterized by invariants under transformations of the projective group. The incidence structure and the cross-ratio are fundamental invariants under projective transformations.
Projective geometry is an elementary non-metrical form of geometry featuring configurations of points and lines (or hyperplanes in higher dimensional spaces) which always meet(!) and which exhibit the principle of duality. Projective geometry can be seen as a geometry of constructions with a straight-edge alone[1]. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels, and no concept of intermediacy[2]. Projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine and Euclidean geometry [3][4]. Projective geometry is not "ordered"[5] and so it is a distinct foundation for geometry. Projective geometry can be modeled by the affine plane (or affine space) plus a line (hyperplane) "at infinity" and then treating that line (or hyperplane) as "ordinary"[6]. An algebraic model for analytic projective geometry is given by homogenous coordinates[7] [8].
Projective geometry was developed by Desargues and others in their exploration of the principles of perspective art[9]. In the early 19th century the work of Poncelet, von Staudt and others established projective geometry as an independent field of mathematics [10]. Its axiomatic foundation was not developed until the work of Gino Fano and Mario Pieri late in the 19th century[11].
