Perspectivity

In geometry and in its applications to drawing, a perspectivity is the formation of an image in a picture plane of a scene viewed from a fixed point. The science of graphical perspective uses perspectivities to make realistic images in proper proportion. According to Kirsti Andersen, the first author to describe perspectivity was Leon Alberti in his De Pictura (1435). In English, Brook Taylor presented his Linear Perspective in 1715, where he explained "Perspective is the Art of drawing on a Plane the Appearances of any Figures, by the Rules of Geometry". In a second book, New Principles of Linear Perspective (1719), Taylor wrote When Lines drawn according to a certain Law from the several Parts of any Figure, cut a Plane, and by that Cutting or Intersection describe a figure on that Plane, that Figure so described is called the Projection of the other Figure. The Lines producing that Projection, taken all together, are called the System of Rays. And when those Rays all pass thro’ one and same Point, they are called the Cone of Rays. And when that Point is consider’d as the Eye of a Spectator, that System of Rays is called the Optic Cone In projective geometry the points of a line are called a projective range, and the set of lines in a plane on a point is called a pencil. Given two lines and in a projective plane and a point P of that plane on neither line, the bijective mapping between the points of the range of and the range of determined by the lines of the pencil on P is called a perspectivity (or more precisely, a central perspectivity with center P). A special symbol has been used to show that points X and Y are related by a perspectivity; In this notation, to show that the center of perspectivity is P, write The existence of a perspectivity means that corresponding points are in perspective. The dual concept, axial perspectivity, is the correspondence between the lines of two pencils determined by a projective range. Projectivity The composition of two perspectivities is, in general, not a perspectivity.

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Perspectivity

Projective range

In mathematics, a projective range is a set of points in projective geometry considered in a unified fashion. A projective range may be a projective line or a conic. A projective range is the dual of a pencil of lines on a given point. For instance, a correlation interchanges the points of a projective range with the lines of a pencil. A projectivity is said to act from one range to another, though the two ranges may coincide as sets. A projective range expresses projective invariance of the relation of projective harmonic conjugates.

Dualité (géométrie projective)

La dualité projective, découverte par Jean-Victor Poncelet, est une généralisation de l'analogie entre le fait que par deux points distincts passe une droite et une seule, et le fait que deux droites distinctes se coupent en un point et un seul (à condition de se placer en géométrie projective, de sorte que deux droites parallèles se rencontrent en un point à l'infini).

Géométrie projective : naissance et fondations

Explore la naissance et les fondements de la géométrie projective, y compris l'invention du point de fuite et la stéréotomie.

Géométrie II : Transformations géométriques modernes

Explore les transformations géométriques et les invariances modernes, en mettant l'accent sur la géométrie projective et les développements historiques.

Ce cours traite des 3 sujets suivants : la perspective, la géométrie descriptive, et une initiation à la géométrie projective.