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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 .
Analytic geometry allowed curves, such as the Folium of Descartes, to be defined using equations instead of geometrical construction. A fundamental advance in the theory of curves was the introduction of analytic geometry by René Descartes in the seventeenth century. This enabled a curve to be described using an equation rather than an ...
In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these ...
Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics.
Over a non-archimedean field analytic geometry is studied via rigid analytic spaces. Modern analytic geometry over the field of complex numbers is closely related to complex algebraic geometry, as has been shown by Jean-Pierre Serre in his paper GAGA, [14] the name of which is French for Algebraic geometry and analytic geometry. The GAGA ...
Secant-, chord-theorem. For the intersecting secants theorem and chord theorem the power of a point plays the role of an invariant: . Intersecting secants theorem: For a point outside a circle and the intersection points , of a secant line with the following statement is true: | | | | = (), hence the product is independent of line .
Projective geometry can be modeled by the affine plane (or affine space) plus a line (hyperplane) "at infinity" and then treating that line (or hyperplane) as "ordinary". [5] An algebraic model for doing projective geometry in the style of analytic geometry is given by homogeneous coordinates.
The vector ♯ is called the sharp of α. The sharp map is a smooth bundle map ♯ : T ∗ M → T M {\displaystyle \sharp :\mathrm {T} ^{*}M\to \mathrm {T} M} . Flat and sharp are mutually inverse isomorphisms of smooth vector bundles, hence, for each p in M , there are mutually inverse vector space isomorphisms between T p M and T ∗