Inmathematics, acubic plane curve is aplane algebraic curveC defined by acubic equation

⁠${\displaystyle F(x,y,z)=0}$⁠

applied tohomogeneous coordinates⁠${\displaystyle (x:y:z)}$⁠ for theprojective plane; or the inhomogeneous version for theaffine space determined by settingz = 1 in such an equation. HereF is a non-zero linear combination of the third-degreemonomials

⁠${\displaystyle x^3,y^3,z^3,x^{2}y,x^{2}z,y^{2}x,y^{2}z,z^{2}x,z^{2}y,xyz}$⁠

These are ten in number; therefore the cubic curves form aprojective space of dimension 9, over any givenfieldK. Each pointP imposes a single linear condition onF, if we ask thatC pass throughP. 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 ingeneral position; compare to two points determining a line and howfive points determine a conic. If two cubics pass through a given set of nine points, then in fact apencil of cubics does, and the points satisfy additional properties; seeCayley–Bacharach theorem.

A cubic curve may have asingular point, in which case it has aparametrization in terms of aprojective line. Otherwise anon-singular cubic curve is known to have nine points ofinflection, over analgebraically closed field such as thecomplex numbers. This can be shown by taking the homogeneous version of theHessian matrix, which defines again a cubic, and intersecting it withC; the intersections are then counted byBézout'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 byIsaac 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 theEuclidean 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 forconic sections, a line cuts this oval at, at most, two points.

A non-singular plane cubic defines anelliptic curve, over any fieldK for which it has a point defined. Elliptic curves are now normally studied in some variant ofWeierstrass's elliptic functions, defining aquadratic extension of the field ofrational functions made by extracting the square root of a cubic. This does depend on having aK-rational point, which serves as thepoint at infinity in Weierstrass form. There are many cubic curves that have no such point, for example whenK is therational number field.

The singular points of an irreducible plane cubic curve are quite limited: onedouble point, or onecusp. A reducible plane cubic curve is either a conic and a line or three lines, and accordingly have two double points or atacnode (if a conic and a line), or up to three double points or a single triple point (concurrent lines) if three lines.

Suppose that△ABC is a triangle with sidelengths${\displaystyle a=|BC|,}$${\displaystyle b=|CA|,}$${\displaystyle c=|AB|.}$ Relative to△ABC, many named cubics pass through well-known points. Examples shown below use two kinds of homogeneous coordinates:trilinear andbarycentric.

To convert from trilinear to barycentric in a cubic equation, substitute as follows:

${\displaystyle x\to bcx,y\to cay,z\to abz;}$

to convert from barycentric to trilinear, use

${\displaystyle x\to ax,y\to by,z\to cz.}$

Many equations for cubics have the form

${\displaystyle f(a,b,c,x,y,z)+f(b,c,a,y,z,x)+f(c,a,b,z,x,y)=0.}$

In the examples below, such equations are written more succinctly in "cyclic sum notation", like this:

${\displaystyle \sum _{\text{cyclic}}f(x,y,z,a,b,c)=0}$.

The cubics listed below can be defined in terms of theisogonal conjugate, denoted byX*, of a pointX not on a sideline of△ABC. A construction ofX* follows. LetL_A be the reflection of lineXA about the internal angle bisector of angleA, and defineL_B andL_C analogously. Then the three reflected lines concur inX*. In trilinear coordinates, if${\displaystyle X=x:y:z,}$ then${\displaystyle X^{\ast }=\frac{1}{x}:\frac{1}{y}:\frac{1}{z}.}$

Trilinear equation:${\displaystyle \sum _{\text{cyclic}}(\cos A-2\cos B\cos C)x(y^2-z^2)=0}$

Barycentric equation:${\displaystyle \sum _{\text{cyclic}}(a^2(b^2+c^2)+(b^2-c^2{)}^2-2a^4)x(c^2{y}^2-b^2{z}^2)=0}$

TheNeuberg cubic (named afterJoseph Jean Baptiste Neuberg) is thelocus of a pointX such thatX* is on the lineEX, whereE is the Euler infinity point (X(30) in theEncyclopedia of Triangle Centers). Also, this cubic is the locus ofX such that the triangle△X_AX_BX_C is perspective to△ABC, where△X_AX_BX_C is the reflection ofX in the linesBC, CA, AB, respectively

The Neuberg cubic passes through the following points:incenter,circumcenter,orthocenter, bothFermat points, bothisodynamic points, the Euler infinity point, other triangle centers, the excenters, the reflections ofA, B, C in the sidelines of△ABC, and the vertices of the six equilateral triangles erected on the sides of△ABC.

For a graphical representation and extensive list of properties of the Neuberg cubic, seeK001 at Berhard Gibert'sCubics in the Triangle Plane.

Trilinear equation:${\displaystyle \sum _{\text{cyclic}}bcx(y^2-z^2)=0}$

Barycentric equation:${\displaystyle \sum _{\text{cyclic}}x(c^2{y}^2-b^2{z}^2)=0}$

The Thomson cubic is the locus of a pointX such thatX* is on the lineGX, whereG is the centroid.

The Thomson cubic passes through the following points: incenter, centroid, circumcenter, orthocenter, symmedian point, other triangle centers, the verticesA, B, C, the excenters, the midpoints of sidesBC, CA, AB, and the midpoints of the altitudes of△ABC. For each pointP on the cubic but not on a sideline of the cubic, the isogonal conjugate ofP is also on the cubic.

For graphs and properties, seeK002 atCubics in the Triangle Plane.

Trilinear equation:${\displaystyle \sum _{\text{cyclic}}(\cos A-\cos B\cos C)x(y^2-z^2)=0}$

Barycentric equation:${\displaystyle \sum _{\text{cyclic}}(2a^2(b^2+c^2)+(b^2-c^2{)}^2-3a^4)x(c^2{y}^2-b^2{z}^2)=0}$

The Darboux cubic is the locus of a pointX such thatX* is on the lineLX, whereL is thede Longchamps point. Also, this cubic is the locus ofX such that the pedal triangle ofX is the cevian triangle of some point (which lies on the Lucas cubic). Also, this cubic is the locus of a pointX such that the pedal triangle ofX and the anticevian triangle ofX are perspective; the perspector lies on the Thomson cubic.

The Darboux cubic passes through the incenter, circumcenter, orthocenter, de Longchamps point, other triangle centers, the verticesA, B, C, the excenters, and the antipodes ofA, B, C on the circumcircle. For each pointP on the cubic but not on a sideline of the cubic, the isogonal conjugate ofP is also on the cubic.

For graphics and properties, seeK004 atCubics in the Triangle Plane.

Trilinear equation:${\displaystyle \sum _{\text{cyclic}}\cos (B-C)x(y^2-z^2)=0}$

Barycentric equation:${\displaystyle \sum _{\text{cyclic}}(a^2(b^2+c^2)+(b^2-c^2{)}^2)x(c^2{y}^2-b^2{z}^2)=0}$

The Napoleon–Feuerbach cubic is the locus of a pointX* is on the lineNX, whereN is the nine-point center, (N =X(5) in theEncyclopedia of Triangle Centers).

The Napoleon–Feuerbach cubic passes through the incenter, circumcenter, orthocenter, 1st and 2nd Napoleon points, other triangle centers, the verticesA, B, C, the excenters, the projections of the centroid on the altitudes, and the centers of the 6 equilateral triangles erected on the sides of△ABC.

For a graphics and properties, seeK005 atCubics in the Triangle Plane.

Trilinear equation:${\displaystyle \sum _{\text{cyclic}}\cos (A)x(b^2{y}^2-c^2{z}^2)=0}$

Barycentric equation:${\displaystyle \sum _{\text{cyclic}}(b^2+c^2-a^2)x(y^2-z^2)=0}$

The Lucas cubic is the locus of a pointX such that the cevian triangle ofX is the pedal triangle of some point; the point lies on the Darboux cubic.

The Lucas cubic passes through the centroid, orthocenter, Gergonne point, Nagel point, de Longchamps point, other triangle centers, the vertices of the anticomplementary triangle, and the foci of the Steiner circumellipse.

For graphics and properties, seeK007 atCubics in the Triangle Plane.

Trilinear equation:${\displaystyle \sum _{\text{cyclic}}bc(a^4-b^2{c}^2)x(y^2+z^2)=0}$

Barycentric equation:${\displaystyle \sum _{\text{cyclic}}(a^4-b^2{c}^2)x(c^2{y}^2+b^2{z}^2)=0}$

Let△A'B'C' be the 1st Brocard triangle. For arbitrary pointX, letX_A, X_B, X_C be the intersections of the linesXA′, XB′, XC′ with the sidelinesBC, CA, AB, respectively. The 1st Brocard cubic is the locus ofX for which the pointsX_A, X_B, X_C are collinear.

The 1st Brocard cubic passes through the centroid, symmedian point, Steiner point, other triangle centers, and the vertices of the 1st and 3rd Brocard triangles.

For graphics and properties, seeK017 atCubics in the Triangle Plane.

Trilinear equation:${\displaystyle \sum _{\text{cyclic}}bc(b^2-c^2)x(y^2+z^2)=0}$

Barycentric equation:${\displaystyle \sum _{\text{cyclic}}(b^2-c^2)x(c^2{y}^2+b^2{z}^2)=0}$

The 2nd Brocard cubic is the locus of a pointX for which the pole of the lineXX* in the circumconic throughX andX* lies on the line of the circumcenter and the symmedian point (i.e., the Brocard axis). The cubic passes through the centroid, symmedian point, both Fermat points, both isodynamic points, the Parry point, other triangle centers, and the vertices of the 2nd and 4th Brocard triangles.

For a graphics and properties, seeK018 atCubics in the Triangle Plane.

Trilinear equation:${\displaystyle \sum _{\text{cyclic}}a(b^2-c^2)x(y^2-z^2)=0}$

Barycentric equation:${\displaystyle \sum _{\text{cyclic}}{a}^2(b^2-c^2)x(c^2{y}^2-b^2{z}^2)=0}$

The 1st equal areas cubic is the locus of a pointX such that area of the cevian triangle ofX equals the area of the cevian triangle ofX*. Also, this cubic is the locus ofX for whichX* is on the lineS*X, whereS is the Steiner point. (S =X(99) in theEncyclopedia of Triangle Centers).

The 1st equal areas cubic passes through the incenter, Steiner point, other triangle centers, the 1st and 2nd Brocard points, and the excenters.

For a graphics and properties, seeK021 atCubics in the Triangle Plane.

Trilinear equation:${\displaystyle (bz+cx)(cx+ay)(ay+bz)=(bx+cy)(cy+az)(az+bx)}$

Barycentric equation:${\displaystyle \sum _{\text{cyclic}}a(a^2-bc)x(c^3{y}^2-b^3{z}^2)=0}$

For any point${\displaystyle X=x:y:z}$ (trilinears), let${\displaystyle X_Y=y:z:x}$ and${\displaystyle X_Z=z:x:y.}$ The 2nd equal areas cubic is the locus ofX such that the area of the cevian triangle ofX_Y equals the area of the cevian triangle ofX_Z.

The 2nd equal areas cubic passes through the incenter, centroid, symmedian point, and points inEncyclopedia of Triangle Centers indexed asX(31),X(105),X(238),X(292),X(365),X(672),X(1453),X(1931),X(2053), and others.

For a graphics and properties, seeK155 atCubics in the Triangle Plane.

• Cayley–Bacharach theorem, on the intersection of two cubic plane curves
• Twisted cubic, a cubic space curve
• Elliptic curve
• Witch of Agnesi
• Catalogue of Triangle Cubics

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• Cubic curves

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• Short description is different from Wikidata