          # Projective geometry

• projective geometry is a topic in mathematics. it is the study of geometric properties that are invariant with respect to projective transformations. this means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. the basic intuitions are that projective space has more points than euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to euclidean points, and vice versa.

properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations (the affine transformations). the first issue for geometers is what kind of geometry is adequate for a novel situation. it is not possible to refer to angles in projective geometry as it is in euclidean geometry, because angle is an example of a concept not invariant with respect to projective transformations, as is seen in perspective drawing. one source for projective geometry was indeed the theory of perspective. another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms. again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. see projective plane for the basics of projective geometry in two dimensions.

while the ideas were available earlier, projective geometry was mainly a development of the 19th century. this included the theory of complex projective space, the coordinates used (homogeneous coordinates) being complex numbers. several major types of more abstract mathematics (including invariant theory, the italian school of algebraic geometry, and felix klein's erlangen programme resulting in the study of the classical groups) were based on projective geometry. it was also a subject with many practitioners for its own sake, as synthetic geometry. another topic that developed from axiomatic studies of projective geometry is finite geometry.

the topic of projective geometry is itself now divided into many research subtopics, 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).

• overview
• history
• description
• duality
• axioms of projective geometry
## Geometry Projecting a sphere to a plane. OutlineHistory Branches Euclidean Non-Euclidean Elliptic Spherical Hyperbolic Non-Archimedean geometry Projective Affine Synthetic Analytic Algebraic Arithmetic Diophantine Differential Riemannian Symplectic Discrete differential Complex Finite Discrete/Combinatorial Digital Convex Computational Fractal Incidence ConceptsFeaturesDimension Straightedge and compass constructions Angle Curve Diagonal Orthogonality (Perpendicular) Parallel Vertex Congruence Similarity Symmetry Zero-dimensional Point One-dimensional Line segment ray Length Two-dimensional Plane Area Polygon Triangle Altitude Hypotenuse Pythagorean theorem Parallelogram Square Rectangle Rhombus Rhomboid Quadrilateral Trapezoid Kite Circle Diameter Circumference Area Three-dimensional Volume Cube cuboid Cylinder Pyramid Sphere Four- / other-dimensional Tesseract Hypersphere Geometers by name Aida Aryabhata Ahmes Alhazen Apollonius Archimedes Atiyah Baudhayana Bolyai Brahmagupta Cartan Coxeter Descartes Euclid Euler Gauss Gromov Hilbert Jyeṣṭhadeva Kātyāyana Khayyám Klein Lobachevsky Manava Minkowski Minggatu Pascal Pythagoras Parameshvara Poincaré Riemann Sakabe Sijzi al-Tusi Veblen Virasena Yang Hui al-Yasamin Zhang List of geometers by period BCE Ahmes Baudhayana Manava Pythagoras Euclid Archimedes Apollonius 1–1400s Zhang Kātyāyana Aryabhata Brahmagupta Virasena Alhazen Sijzi Khayyám al-Yasamin al-Tusi Yang Hui Parameshvara 1400s–1700s Jyeṣṭhadeva Descartes Pascal Minggatu Euler Sakabe Aida 1700s–1900s Gauss Lobachevsky Bolyai Riemann Klein Poincaré Hilbert Minkowski Cartan Veblen Coxeter Present day Atiyah Gromov vt Projective geometry is a topic in mathematics. It is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice versa. Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations (the affine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. It is not possible to refer to angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant with respect to projective transformations, as is seen in perspective drawing. One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of projective geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century. This included the theory of complex projective space, the coordinates used (homogeneous coordinates) being complex numbers. Several major types of more abstract mathematics (including invariant theory, the Italian school of algebraic geometry, and Felix Klein's Erlangen programme resulting in the study of the classical groups) were based on projective geometry. It was also a subject with many practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry. The topic of projective geometry is itself now divided into many research subtopics, 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). Contents 1 Overview 2 History 3 Description 4 Duality 5 Axioms of projective geometry 5.1 Whitehead's axioms 5.2 Additional axioms 5.3 Axioms using a ternary relation 5.4 Axioms for projective planes 6 See also 7 Notes 8 References 9 External links  