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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 lines, 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 wikiPageRevisionID "594018964".
- Cubic_plane_curve hasPhotoCollection Cubic_plane_curve.
- Cubic_plane_curve subject Category:Algebraic_curves.
- Cubic_plane_curve type Abstraction100002137.
- Cubic_plane_curve type AlgebraicCurves.
- Cubic_plane_curve type Attribute100024264.
- Cubic_plane_curve type Curve113867641.
- Cubic_plane_curve type Line113863771.
- Cubic_plane_curve type Shape100027807.
- 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 "Courbe cubique".
- Cubic_plane_curve label "Cubic plane curve".
- Cubic_plane_curve label "Кубика".
- Cubic_plane_curve sameAs Courbe_cubique.
- Cubic_plane_curve sameAs m.02_j4t.
- Cubic_plane_curve sameAs Q2369721.
- Cubic_plane_curve sameAs Q2369721.
- Cubic_plane_curve sameAs Cubic_plane_curve.
- Cubic_plane_curve wasDerivedFrom Cubic_plane_curve?oldid=594018964.
- Cubic_plane_curve depiction CubicCurve.svg.
- Cubic_plane_curve isPrimaryTopicOf Cubic_plane_curve.