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Geometry. 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.
Projective space. In graphical perspective, parallel (horizontal) lines in the plane intersect at a vanishing point (on the horizon). In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus be viewed as the extension of a ...
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the pole or center of projection), onto a plane (the projection plane) perpendicular to the diameter through the point. It is a smooth, bijective function from the entire sphere except the center of projection to the ...
In projective geometry, the sphere is an example of a complex projective space and can be thought of as the complex projective line (), the projective space of all complex lines in . As with any compact Riemann surface, the sphere may also be viewed as a projective algebraic curve , making it a fundamental example in algebraic geometry .
Construction. Complex projective space is a complex manifold that may be described by n + 1 complex coordinates as. where the tuples differing by an overall rescaling are identified: That is, these are homogeneous coordinates in the traditional sense of projective geometry. The point set CPn is covered by the patches .
In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry . Analytic geometry is used in physics and engineering , and also in aviation , rocketry , space science , and spaceflight .
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, [1][2][3] 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 ...
In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by points and lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one through language (§ Principle of duality) and the other a more functional approach through special ...