Foundations of Projective Geometry. Questo articolo non è disponibile.Foundations of Projective Geometry

The first geometrical properties of a projective nature were discovered in the third century by Pappus of Alexandria. Filippo Brunelleschi (1404-1472) started investigating the geometry of perspective in 1425. Johannes Kepler (1571-1630) and Gerard Desargues (1591-1661) independently developed the pivotal concept of the "point at infinity". Desargues developed 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. Desargues's study on conic sections drew the attention of 16-years old Blaise Pascal and helped him formulate Pascal's theorem. The works of Gaspard Monge at the end of 18th and beginning of 19th century were important for the subsequent development of projective geometry. The work of Desargues was 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. Poncelet separated the projective properties of objects in individual class and establishing a relationship between metric and projective properties. The non-Euclidean geometries discovered shortly thereafter were eventually demonstrated to have models, such as the Klein model of hyperbolic space, relating to projective geometry.

Robin Hartshorne is a professor of mathematics, emeritus, at the University of California at Berkeley. His area of research is algebraic geometry, about which he has written a textbook and many research articles. He also has a life-long interest in elementary geometry, starting from his class in mechanical drawing in the fifth grade of elementary school. Recently, based on a course he taught at Berkeley, he wrote a text of Euclidean and non-Euclidean geometry, called Geometry, Euclid and Beyond. Aside from mathematics, he is a musician (flute, piano, shakuhachi), and an avid mountain climber. He also likes languages, speaking fluent English, French, German and Japanese. His wife, Edie, is a therapist, musician, and author. They have two children, Ben, a computer systems engineer, and Joemy, a third grade elementary school teacher.