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- Generalized_conic abstract "In mathematics, a generalized conic is a geometrical object defined by a property which is a generalization of some defining property of the classical conic. For example, in elementary geometry, an ellipse can be defined as the locus of a point which moves in a plane such that the sum of its distances from two fixed points in the plane is a constant. The curve obtained when the set of two fixed points is replaced by an arbitrary, but fixed, finite set of points in the plane can be thought of as a generalized ellipse. Since an ellipse is the equidistant set of two circles, the equidistant set of two arbitrary sets of points in a plane can be viewed as a generalized conic. In rectangular Cartesian coordinates, the equation y = x2 represents a parabola. The generalized equation y = xr, for r ≠ 0 and r ≠ 1, can be treated as defining a generalized parabola. The idea of generalized conic has found applications in approximation theory and optimization theory.Among the several possible ways in which the concept of a conic can be generalized, the most widely used approach is to define it as a generalization of the ellipse. The starting point for this approach is to look upon an ellipse as a curve satisfying the 'two-focus property': an ellipse is a curve that is the locus of points the sum of whose distances from two given points is constant. The two points are the foci of the ellipse. The curve obtained by replacing the set of two fixed points by an arbitrary, but fixed, finite set of points in the plane can be thought of as a generalized ellipse. Generalized conics with three foci are called trifocal ellipses. This can be further generalized to curves which are obtained as the loci of points which move such that the some of weighted arithmetic mean of the distances from a finite set of points is a constant. A still further generalization is possible by assuming that the weights attached to the distances can be of arbitrary sign, namely, plus or minus. Finally, the restriction that the set of fixed points, called the set of foci of the generalized conic, be finite may also be removed. The set may be assumed to be finite or infinite. In the infinite case, the weighted arithmetic mean has to be replaced by an appropriate integral. Generalized conics in this sense are also called polyellipses, egglipses, or, generalized ellipses. Since such curves were considered by the German mathematician Ehrenfried Walther von Tschirnhaus (1651 – 1708) they are also known as Tschirnhaus'sche Eikurve. Also such generalizations have been discussed by Rene Descartes and by James Clerk Maxwell.".
- Generalized_conic thumbnail Construction_of_bifocal_oval_by_Maxwell.jpg?width=300.
- Generalized_conic wikiPageID "48877966".
- Generalized_conic wikiPageLength "20079".
- Generalized_conic wikiPageOutDegree "45".
- Generalized_conic wikiPageRevisionID "701713317".
- Generalized_conic wikiPageWikiLink Approximation_theory.
- Generalized_conic wikiPageWikiLink Cardinality.
- Generalized_conic wikiPageWikiLink Category:Algebraic_curves.
- Generalized_conic wikiPageWikiLink Category:Conic_sections.
- Generalized_conic wikiPageWikiLink Conic_section.
- Generalized_conic wikiPageWikiLink Eccentricity_(mathematics).
- Generalized_conic wikiPageWikiLink Ehrenfried_Walther_von_Tschirnhaus.
- Generalized_conic wikiPageWikiLink Ellipse.
- Generalized_conic wikiPageWikiLink Empty_set.
- Generalized_conic wikiPageWikiLink Equidistant_set.
- Generalized_conic wikiPageWikiLink File:Construction_of_bifocal_oval_by_Maxwell.jpg.
- Generalized_conic wikiPageWikiLink File:Construction_of_trifocal_oval_by_Maxwell.jpg.
- Generalized_conic wikiPageWikiLink File:Equidistant-set-of-two-circles-ellipse.gif.
- Generalized_conic wikiPageWikiLink File:Generalized_conic_01.png.
- Generalized_conic wikiPageWikiLink File:Generalized_conic_02.png.
- Generalized_conic wikiPageWikiLink File:Generalized_conic_03.png.
- Generalized_conic wikiPageWikiLink File:Generalized_conic_04.png.
- Generalized_conic wikiPageWikiLink File:Generalized_conic_05.png.
- Generalized_conic wikiPageWikiLink File:Generalized_conic_06.png.
- Generalized_conic wikiPageWikiLink File:Generalized_conic_07.png.
- Generalized_conic wikiPageWikiLink File:Generalized_conic_08.png.
- Generalized_conic wikiPageWikiLink File:Unwrapping_a_cone_01.jpg.
- Generalized_conic wikiPageWikiLink File:Unwrapping_a_cone_02.jpg.
- Generalized_conic wikiPageWikiLink Focus_(geometry).
- Generalized_conic wikiPageWikiLink Generalization.
- Generalized_conic wikiPageWikiLink Geometry.
- Generalized_conic wikiPageWikiLink James_Clerk_Maxwell.
- Generalized_conic wikiPageWikiLink Lemniscate.
- Generalized_conic wikiPageWikiLink Locus_(mathematics).
- Generalized_conic wikiPageWikiLink Mamikon_Mnatsakanian.
- Generalized_conic wikiPageWikiLink Mathematical_optimization.
- Generalized_conic wikiPageWikiLink Mathematics.
- Generalized_conic wikiPageWikiLink Metric_space.
- Generalized_conic wikiPageWikiLink Polar_coordinate_system.
- Generalized_conic wikiPageWikiLink René_Descartes.
- Generalized_conic wikiPageWikiLink Taxicab_geometry.
- Generalized_conic wikiPageWikiLink Tom_M._Apostol.
- Generalized_conic wikiPageWikiLink Tomography.
- Generalized_conic wikiPageWikiLink Weighted_arithmetic_mean.
- Generalized_conic wikiPageWikiLinkText "Generalized conic".
- Generalized_conic wikiPageWikiLinkText "generalized conic".
- Generalized_conic wikiPageUsesTemplate Template:Reflist.
- Generalized_conic subject Category:Algebraic_curves.
- Generalized_conic subject Category:Conic_sections.
- Generalized_conic hypernym Object.
- Generalized_conic type Planet.
- Generalized_conic comment "In mathematics, a generalized conic is a geometrical object defined by a property which is a generalization of some defining property of the classical conic. For example, in elementary geometry, an ellipse can be defined as the locus of a point which moves in a plane such that the sum of its distances from two fixed points in the plane is a constant.".
- Generalized_conic label "Generalized conic".
- Generalized_conic wasDerivedFrom Generalized_conic?oldid=701713317.
- Generalized_conic depiction Construction_of_bifocal_oval_by_Maxwell.jpg.
- Generalized_conic isPrimaryTopicOf Generalized_conic.