In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean 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 motivated by projective geometry.
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Ce cours entend exposer les fondements de la géométrie à un triple titre :
1/ de technique mathématique essentielle au processus de conception du projet,
2/ d'objet privilégié des logiciels de concept
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into a logical system in which each result is proved from axioms and previously proved theorems.
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.
Linear algebra is the branch of mathematics concerning linear equations such as: linear maps such as: and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.
In geometry, Brianchon's theorem is a theorem stating that when a hexagon is circumscribed around a conic section, its principal diagonals (those connecting opposite vertices) meet in a single point. It is named after Charles Julien Brianchon (1783–1864). Let be a hexagon formed by six tangent lines of a conic section. Then lines (extended diagonals each connecting opposite vertices) intersect at a single point , the Brianchon point. The polar reciprocal and projective dual of this theorem give Pascal's theorem.
In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An incidence structure is what is obtained when all other concepts are removed and all that remains is the data about which points lie on which lines. Even with this severe limitation, theorems can be proved and interesting facts emerge concerning this structure.
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts.
In the present thesis, we delve into different extremal and algebraic problems arising from combinatorial geometry. Specifically, we consider the following problems. For any integer n≥3, we define e(n) to be the minimum positive integer such that any set of e(n) points in general position in the plane contains n points in convex position. In 1935, Erd\H{o}s and Szekeres proved that e(n)≤(n−22n−4)+1 and later in 1961, they obtained the lower bound 2n−2+1≤e(n), which they conjectured to be optimal. We prove that e(n)≤(n−22n−5)−(n−32n−8)+2. In a recent breakthrough, Suk proved that e(n)≤2n+O(n2/3logn). We strengthen this result by extending it to pseudo-configurations and also improving the error term. Combining our results with a theorem of Dobbins et al., we significantly improve the best known upper bounds on the following two functions, introduced by Bisztriczky and Fejes T'{o}th and by Pach and T'{o}th, respectively. Let c(n) (and c′(n)) denote the smallest positive integer N such that any family of N pairwise disjoint convex bodies in general position (resp., N convex bodies in general position, any pair of which share at most two boundary points) has an n members in convex position. We show that c(n)≤c′(n)≤2n+O(nlogn). Given a point set P in the plane, an ordinary circle for P is defined as a circle containing exactly three points of P. We prove that any set of n points in the plane, not all on a line or a circle, determines at least 41n2−O(n) ordinary circles. We determine the exact minimum number of ordinary circles for all sufficiently large n, and characterize all point sets that come close to this minimum. We also consider the orchard problem for circles, where we determine the maximum number of circles containing four points of a given set and describe the extremal configurations. A special case of the Schwartz-Zippel lemma states that given an algebraic curve C⊂C2 of degree d and two finite sets A,B⊂C, we have ∣C∩(A×B)∣=Od(∣A∣+∣B∣). We establish a two-dimensional version of this result, and prove upper bounds on the size of the intersection ∣X∩(P×Q)∣ for a variety X⊂C4 and finite sets P,Q⊂C2. A key ingredient in our proofs is a two-dimensional version of a special case of Alon's combinatorial Nullstellensatz. As corollaries, we generalize the Szemer'edi-Trotter point-line incidence theorem and several known bounds on repeated and distinct Euclidean distances. We use incidence geometry to prove some sum-product bounds over arbitrary fields. First, we give an explicit exponent and improve a recent result of Bukh and Tsimerman by proving that max{∣A+A∣,∣f(A,A)∣}≫∣A∣6/5 for any small set A⊂Fp and quadratic non-degenerate polynomial f(x,y)∈Fp[x,y]. This generalizes the result of Roche-Newton et al. giving the best known lower bound for the term max{∣A+A∣,∣A⋅A∣}. Secondly, we improve and generalize the sum-product results of Hegyv'{a}ri and Hennecart on max{∣A+B∣,∣f(B,C)∣}, for a specific type of function f. Finally, we prove that the number of distinct cubic distances generated by any small set A×A⊂Fp2 is Ω(∣A∣8/7), which improves a result of Yazici et al.
