quadratic curves (original) (raw)

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

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

If C≠0, we will rotate the coordinate system, getting new coordinate axes x′ and y′, such that the equation (1) transforms into a new one having no more the mixed term x′⁢y′. Let the rotation 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 matrix). Substituting these expressions into (1) it becomes

where

It’s always possible to determine α such that (B-A)⁢sin⁡2⁢α=-2⁢C⁢cos⁡2⁢α, i.e. that

for A≠B and α=45∘ for the case A=B. Then the term 2⁢P⁢x′⁢y′ vanishes in (2), which becomes, dropping out the apostrophes,

• •
  If none of the coefficients M and M equal zero, one can remove the first degree terms of (4) by first writing it as

  M⁢(x+GM)2+N⁢(y+HN)2=G2M+H2N-F
  and then translating the origin to the point (-GM,-HN) , when we obtain the equation of the form
  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 ellipse (http://planetmath.org/Ellipse2)
  If M and N have opposite signs and K≠0, then the curve (5) correspondingly is one of the hyperbolas (http://planetmath.org/Hyperbola2)
  x2(|K/M
  ----------------------------
  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
  i.e.

  M⁢(x+GM)2+2⁢H⁢(y+M⁢F-G22⁢H⁢M)=0.
  Translating now the origin to the point (-GM,G2-M⁢F2⁢H⁢M) the equation changes to
  For H≠0, this is the equation y=-M2⁢H⁢x2 of a parabola, but for H=0, of a double line x2=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 connection

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

Thus the curve A⁢x2+B⁢y2+2⁢C⁢x⁢y+2⁢D⁢x+2⁢E⁢y+F=0 is, when it is real,

1. 1. for A⁢B-C2>0 an ellipse (http://planetmath.org/Ellipse2),
2. 1. for A⁢B-C2<0 a hyperbola (http://planetmath.org/Hyperbola2) or two intersecting lines,
3. 1. for A⁢B-C2=0 a parabola (http://planetmath.org/Parabola2) or a double line.

References

• 1 L. Lindelöf: Analyyttisen geometrian oppikirja. Kolmas painos. Suomalaisen Kirjallisuuden Seura, Helsinki (1924).