Definition:Projective Geometry
Definition
Projective geometry is the field of geometry concerned with properties which are invariant under projective transformations.
Also see
- Results about projective geometry can be found here.
Historical Note
The study of projective geometry was pioneered in $1639$ by Girard Desargues, as a result of the work he was doing with conic sections.
He was influenced by perspective in art, and particularly interested in how a projection of a conic section is itself a conic section.
He made the assumption that parallel lines meet at a point at infinity.
Like Johannes Kepler, he considered the parabola to have a second focus at infinity.
He studied properties of conic sections which were unchanged under projections.
He also made use of the complete quadrangle because of its harmonic ratios.
Projective geometry was neglected at the time, but it did inspire Blaise Pascal in his early work.
However, it was revived and grew to considerable importance in the $19$th century, mainly as the result of work by Jean-Victor Poncelet.
Sources
- 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{V}$: "Greatness and Misery of Man"
- 1952: T. Ewan Faulkner: Projective Geometry (2nd ed.) ... (previous) ... (next): Chapter $1$: Introduction: The Propositions of Incidence: $1.1$: Historical Note
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): geometry
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): projective geometry
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): geometry
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): projective geometry
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): projective geometry