Jean-Victor Poncelet (ʒɑ̃ viktɔʁ pɔ̃slɛ; 1 July 1788 – 22 December 1867) was a French engineer and mathematician who served most notably as the Commanding General of the École Polytechnique. He is considered a reviver of projective geometry, and his work Traité des propriétés projectives des figures is considered the first definitive text on the subject since Gérard Desargues' work on it in the 17th century. He later wrote an introduction to it: Applications d'analyse et de géométrie.
As a mathematician, his most notable work was in projective geometry, although an early collaboration with Charles Julien Brianchon provided a significant contribution to Feuerbach's theorem. He also made discoveries about projective harmonic conjugates; relating these to the poles and polar lines associated with conic sections. He developed the concept of parallel lines meeting at a point at infinity and defined the circular points at infinity that are on every circle of the plane. These discoveries led to the principle of duality, and the principle of continuity and also aided in the development of complex numbers.
As a military engineer, he served in Napoleon's campaign against the Russian Empire in 1812, in which he was captured and held prisoner until 1814. Later, he served as a professor of mechanics at the École d'application in his home town of Metz, during which time he published Introduction à la mécanique industrielle, a work he is famous for, and improved the design of turbines and water wheels. In 1837, a tenured 'Chaire de mécanique physique et expérimentale' was specially created for him at the Sorbonne (the University of Paris). In 1848, he became the commanding general of his alma mater, the École Polytechnique. He is honoured by having his name listed among notable French engineers and scientists displayed around the first stage of the Eiffel tower.
Poncelet was born in Metz, France, on 1 July 1788, the illegitimate then legitimated son of Claude Poncelet, a lawyer of the Parliament of Metz and wealthy landowner.
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A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties. The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions.
In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept. There are two approaches to the subject of duality, one through language () and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration.
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