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- Cubic_plane_curve abstract "In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equationF(x,y,z) = 0applied to homogeneous coordinates x:y:z for the projective plane; or the inhomogeneous version for the affine space determined by setting z = 1 in such an equation. Here F is a non-zero linear combination of the third-degree monomials x3, y3, z3, x2y, x2z, y2x, y2z, z2x, z2y, xyz.These are ten in number; therefore the cubic curves form a projective space of dimension 9, over any given field K. Each point P imposes a single linear condition on F, if we ask that C pass through P. Therefore we can find some cubic curve through any nine given points, which may be degenerate, and may not be unique, but will be unique and non-degenerate if the points are in general position; compare to two points determining a line and how five points determine a conic. If two cubics pass through a given set of nine points, then in fact a pencil of cubics does, and the points satisfy additional properties; see Cayley–Bacharach theorem.A cubic curve may have a singular point, in which case it has a parametrization in terms of a projective line. Otherwise a non-singular cubic curve is known to have nine points of inflection, over an algebraically closed field such as the complex numbers. This can be shown by taking the homogeneous version of the Hessian matrix, which defines again a cubic, and intersecting it with C; the intersections are then counted by Bézout's theorem. However, only three of these points may be real, so that the others cannot be seen in the real projective plane by drawing the curve. The nine inflection points of a non-singular cubic have the property that every line passing through two of them contains exactly three inflection points.The real points of cubic curves were studied by Isaac Newton. The real points of a non-singular projective cubic fall into one or two 'ovals'. One of these ovals crosses every real projective line, and thus is never bounded when the cubic is drawn in the Euclidean plane; it appears as one or three infinite branches, containing the three real inflection points. The other oval, if it exists, does not contain any real inflection point and appears either as an oval or as two infinite branches. Like for conic sections, a line cuts this oval at, at most, two points.A non-singular cubic defines an elliptic curve, over any field K for which it has a point defined. Elliptic curves are now normally studied in some variant of Weierstrass's elliptic functions, defining a quadratic extension of the field of rational functions made by extracting the square root of a cubic. This does depend on having a K-rational point, which serves as the point at infinity in Weierstrass form. There are many cubic curves that have no such point, for example when K is the rational number field.The singular points of an irreducible plane cubic curve are quite limited: one double point, or one cusp. A reducible plane cubic curve is either a conic and a line or three lines, and accordingly have two double points or a tacnode (if a conic and a line), or up to three double points or a single triple point (concurrent lines) if three lines.".
- Cubic_plane_curve thumbnail CubicCurve.svg?width=300.
- Cubic_plane_curve wikiPageExternalLink Intro&Zcubics.html.
- Cubic_plane_curve wikiPageExternalLink k001.html.
- Cubic_plane_curve wikiPageExternalLink k002.html.
- Cubic_plane_curve wikiPageExternalLink k004.html.
- Cubic_plane_curve wikiPageExternalLink k005.html.
- Cubic_plane_curve wikiPageExternalLink k007.html.
- Cubic_plane_curve wikiPageExternalLink k017.html.
- Cubic_plane_curve wikiPageExternalLink k018.html.
- Cubic_plane_curve wikiPageExternalLink k021.html.
- Cubic_plane_curve wikiPageExternalLink k155.html.
- Cubic_plane_curve wikiPageExternalLink isocubics.html.
- Cubic_plane_curve wikiPageExternalLink index.html.
- Cubic_plane_curve wikiPageExternalLink cubics.htm.
- Cubic_plane_curve wikiPageExternalLink cubics.htm.
- Cubic_plane_curve wikiPageID "649721".
- Cubic_plane_curve wikiPageLength "18361".
- Cubic_plane_curve wikiPageOutDegree "54".
- Cubic_plane_curve wikiPageRevisionID "680095058".
- Cubic_plane_curve wikiPageWikiLink Affine_space.
- Cubic_plane_curve wikiPageWikiLink Algebraic_curve.
- Cubic_plane_curve wikiPageWikiLink Algebraically_closed_field.
- Cubic_plane_curve wikiPageWikiLink Altitude_(triangle).
- Cubic_plane_curve wikiPageWikiLink Barycentric_coordinate_system.
- Cubic_plane_curve wikiPageWikiLink Bxc3xa9zouts_theorem.
- Cubic_plane_curve wikiPageWikiLink Category:Algebraic_curves.
- Cubic_plane_curve wikiPageWikiLink Cayley–Bacharach_theorem.
- Cubic_plane_curve wikiPageWikiLink Circumscribed_circle.
- Cubic_plane_curve wikiPageWikiLink Complex_number.
- Cubic_plane_curve wikiPageWikiLink Concurrent_lines.
- Cubic_plane_curve wikiPageWikiLink Conic_section.
- Cubic_plane_curve wikiPageWikiLink Cusp_(singularity).
- Cubic_plane_curve wikiPageWikiLink De_Longchamps_point.
- Cubic_plane_curve wikiPageWikiLink Elliptic_curve.
- Cubic_plane_curve wikiPageWikiLink Encyclopedia_of_Triangle_Centers.
- Cubic_plane_curve wikiPageWikiLink Fermat_point.
- Cubic_plane_curve wikiPageWikiLink Field_(mathematics).
- Cubic_plane_curve wikiPageWikiLink Five_points_determine_a_conic.
- Cubic_plane_curve wikiPageWikiLink General_position.
- Cubic_plane_curve wikiPageWikiLink Hessian_matrix.
- Cubic_plane_curve wikiPageWikiLink Homogeneous_coordinates.
- Cubic_plane_curve wikiPageWikiLink Incenter.
- Cubic_plane_curve wikiPageWikiLink Inflection_point.
- Cubic_plane_curve wikiPageWikiLink Isaac_Newton.
- Cubic_plane_curve wikiPageWikiLink Isodynamic_point.
- Cubic_plane_curve wikiPageWikiLink Joseph_Jean_Baptiste_Neuberg.
- Cubic_plane_curve wikiPageWikiLink Kummer_theory.
- Cubic_plane_curve wikiPageWikiLink Locus_(mathematics).
- Cubic_plane_curve wikiPageWikiLink Mathematics.
- Cubic_plane_curve wikiPageWikiLink Monomial.
- Cubic_plane_curve wikiPageWikiLink Pencil_(mathematics).
- Cubic_plane_curve wikiPageWikiLink Point_at_infinity.
- Cubic_plane_curve wikiPageWikiLink Projective_line.
- Cubic_plane_curve wikiPageWikiLink Projective_plane.
- Cubic_plane_curve wikiPageWikiLink Projective_space.
- Cubic_plane_curve wikiPageWikiLink Rational_function.
- Cubic_plane_curve wikiPageWikiLink Rational_number.
- Cubic_plane_curve wikiPageWikiLink Rational_point.
- Cubic_plane_curve wikiPageWikiLink Singular_point_of_a_curve.
- Cubic_plane_curve wikiPageWikiLink Singularity_(mathematics).
- Cubic_plane_curve wikiPageWikiLink Tacnode.
- Cubic_plane_curve wikiPageWikiLink Trilinear_coordinates.
- Cubic_plane_curve wikiPageWikiLink Twisted_cubic.
- Cubic_plane_curve wikiPageWikiLink Two-dimensional_space.
- Cubic_plane_curve wikiPageWikiLink Weierstrasss_elliptic_functions.
- Cubic_plane_curve wikiPageWikiLink File:CubicCurve.svg.
- Cubic_plane_curve wikiPageWikiLink File:Cubic_with_double_point.svg.
- Cubic_plane_curve wikiPageWikiLink File:Thomson_cubic.svg.
- Cubic_plane_curve wikiPageWikiLinkText "Cubic plane curve – Thomson cubic".
- Cubic_plane_curve wikiPageWikiLinkText "Cubic plane curve".
- Cubic_plane_curve wikiPageWikiLinkText "Cubic plane curve#Darboux cubic".
- Cubic_plane_curve wikiPageWikiLinkText "Cubic with double point".
- Cubic_plane_curve wikiPageWikiLinkText "Neuberg cubic".
- Cubic_plane_curve wikiPageWikiLinkText "cubic curve".
- Cubic_plane_curve wikiPageWikiLinkText "cubic curves in the plane".
- Cubic_plane_curve wikiPageWikiLinkText "cubic curves".
- Cubic_plane_curve wikiPageWikiLinkText "cubic plane curve".
- Cubic_plane_curve wikiPageWikiLinkText "cubic plane".
- Cubic_plane_curve wikiPageWikiLinkText "cubic".
- Cubic_plane_curve wikiPageUsesTemplate Template:=.
- Cubic_plane_curve wikiPageUsesTemplate Template:Citation.
- Cubic_plane_curve wikiPageUsesTemplate Template:Redirect.
- Cubic_plane_curve subject Category:Algebraic_curves.
- Cubic_plane_curve hypernym C.
- Cubic_plane_curve type SoccerClubSeason.
- Cubic_plane_curve type Variety.
- Cubic_plane_curve comment "In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equationF(x,y,z) = 0applied to homogeneous coordinates x:y:z for the projective plane; or the inhomogeneous version for the affine space determined by setting z = 1 in such an equation.".
- Cubic_plane_curve label "Cubic plane curve".
- Cubic_plane_curve sameAs Q2369721.
- Cubic_plane_curve sameAs Kolmannen_asteen_käyrä.
- Cubic_plane_curve sameAs Courbe_cubique.
- Cubic_plane_curve sameAs m.02_j4t.
- Cubic_plane_curve sameAs Кубика.
- Cubic_plane_curve sameAs Krivulja_tretje_stopnje.
- Cubic_plane_curve sameAs Đường_cong_bậc_ba_Neuberg.
- Cubic_plane_curve sameAs Q2369721.
- Cubic_plane_curve wasDerivedFrom Cubic_plane_curve?oldid=680095058.
- Cubic_plane_curve depiction CubicCurve.svg.
- Cubic_plane_curve isPrimaryTopicOf Cubic_plane_curve.