Archive for December, 2008
Inversion is a transformation of the Euclidean plane closely related to circles. It is seldom taught in high schools (at least in France), but it has special interest when dealing with circles, since it maps circles and lines to circles or lines. Inversion is defined with respect to a center O and a radius ρ (in French, the inversion is said to have pôle O and puissance (power) ρ²).
Euclidean geometry is closely related to the ability to define circles, which exists naturally in two settings: the complex projective line and the real projective plane.
The Riemann sphere
The complex projective line is a natural compactification of the Euclidean plane: its natural automorphisms are homographies or Möbius transformations, which transform lines or circles into lines or circles. The Euclidean geometry arises from the natural isomorphism between the group of homographies stabilizing the point at infinity and the group of direct isometries of the plane .
Peter Woit, the author of the blog Not Even Wrong, is writing a series of articles about BRST cohomology and Lie algebra cohomology, as pointed out by the Secret Blogging seminar. Since these theories are now fundamental in the study of gauge symmetry in quantum mechanics and Lie group actions on differentiable manifolds (these subjects now being often blended into each other), having a nice introduction written as blog posts is undoubtedly useful !
John Casey’s Treatise on the analytical geometry of the point, line, circle and conic sections is a book from the end of the 19th century gathering a wide range of results and constructions in plane geometry known at those times. The Bibliothèque Nationale de France gives free access to a digitized version of the book (see there).
The book begins with the definitions of classical coordinate systems: cartesian coordinates, polar coordinates, projective homogeneous coordinates (which were then dubbed trilinear or barycentric). Casey has no taboo concerning points of the field of complex numbers: in particular, many statements make use of cyclic points or zero-radius circles. Much further into the book, the author introduces projections and perspectives, as well as quadratic cones, and harmonic divisions.
Commençons par un peu de géométrie élémentaire… Soit un triangle non dégénéré dans le plan euclidien. L’usage de coordonnées barycentriques permet de repérer un point du plan par trois nombres réels de sorte que soit le barycentre des sommets du triangle avec les poids indiqués.
Let’s begin with elementary geometry… Let be a non-degenerate triangle in the Euclidean plane. By using barycentric coordinates, any point in the plan can be assigned a triple of real numbers , such that is the barycentre of the vertices with these weights.
Ainsi s’ouvrent mes premiers pas dans les technologies inconnues qui hantent notre monde. À travers ce tissu de bûches, j’espère parler de mathématiques, et peut-être d’autres choses, en français ou en d’autres langues.