quadratic curves

We want to determine the graphical representant of the general bivariate quadratic equation

Ax2+By2+2Cxy+2Dx+2Ey+F=0, (1)

where A,B,C,D,E,F are known real numbers and  A2+B2+C2>0.

If  C0, we will rotate the coordinate systemMathworldPlanetmath, getting new coordinateMathworldPlanetmathPlanetmath axes x and y, such that the equation (1) transforms into a new one having no more the mixed termPlanetmathPlanetmath xy.  Let the rotationMathworldPlanetmath angle be α to the anticlockwise (positive) direction so that the x- and y-axes form the angles α and α+90 with the original x-axis, respectively.  Then there is the


between the new and old coordinates (see rotation matrixMathworldPlanetmath).  Substituting these expressions into (1) it becomes

Mx2+Ny2+2Pxy+2Gx+2Hy+F=0, (2)


{M=Acos2α+Bsin2α+Csin2α,N=Asin2α+Bcos2α-Csin2α,2P=(B-A)sin2α+2Ccos2α. (3)

It’s always possible to determine α such that  (B-A)sin2α=-2Ccos2α,  i.e. that


for  AB  and  α=45  for the case  A=B.  Then the term 2Pxy vanishes in (2), which becomes, dropping out the apostrophes,

Mx2+Ny2+2Gx+2Hy+F=0. (4)
  • If none of the coefficients M and M equal zero, one can remove the first degree terms of (4) by first writing it as


    and then translating the origin to the point  (-GM,-HN) ,  when we obtain the equation of the form

    Mx2+Ny2=K. (5)

    If M and M have the same sign (http://planetmath.org/SignumFunction), then in that (5) could have a counterpart in the plane, the sign must be the same as the sign of K; then the counterpart is the ellipsePlanetmathPlanetmath (http://planetmath.org/Ellipse2)


    If M and N have opposite signs and  K0,  then the curve (5) correspondingly is one of the hyperbolas (http://planetmath.org/Hyperbola2)


    which for  K=0  is reduced to a pair of intersecting lines.

  • If one of M and N, e.g. the latter, is zero, the equation (4) may be written




    Translating now the origin to the point  (-GM,G2-MF2HM)  the equation changes to

    Mx2+2Hy=0. (6)

    For  H0,  this is the equation  y=-M2Hx2  of a parabola, but for  H=0,  of a double linex2=0.

The kind of the quadratic curve (1) can also be found out directly from this original form of the equation. Namely, from the formulae (3) between the old and the new coefficients one may derive the connectionMathworldPlanetmath

MN-P2=AB-C2 (7)

when one first adds and subtracts them obtaining


Two latter of these give


and when one subtracts this from the equation  (M+N)2=(A+B)2,  the result is (7), which due to the choice of α is simply

MN=AB-C2. (8)

Thus the curve  Ax2+By2+2Cxy+2Dx+2Ey+F=0  is, when it is real,

  1. 1.

    for  AB-C2>0  an ellipse (http://planetmath.org/Ellipse2),

  2. 2.

    for  AB-C2<0  a hyperbola (http://planetmath.org/Hyperbola2) or two intersecting lines,

  3. 3.

    for  AB-C2=0  a parabola (http://planetmath.org/Parabola2) or a double line.


  • 1 L. Lindelöf: Analyyttisen geometrian oppikirja.  Kolmas painos.  Suomalaisen Kirjallisuuden Seura, Helsinki (1924).
Title quadratic curves
Canonical name QuadraticCurves
Date of creation 2013-03-22 17:56:26
Last modified on 2013-03-22 17:56:26
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Topic
Classification msc 51N20
Synonym graph of quadratic equation
Related topic ConicSection
Related topic TangentOfConicSection
Related topic OsculatingCurve
Related topic IntersectionOfQuadraticSurfaceAndPlane
Related topic PencilOfConics
Related topic SimplestCommonEquationOfConics
Defines double line