What is the graph of cubic equation..

Question

I know that a parabola is the locus of the points found at equal distance from the focus and the directrix.

Is there any similar geometric description of cubic curves, that would also allow me to draw their graphs?

Answer 1

I assume that by “quadratic equation” you are referring to something like

$$y=ax^2+bx+c$$

or some such. The equation of a parabola looks like this if the directrix of the parabola is horizontal. If not, the equation becomes more complicated. In general you write something like $$ax^2+bxy+cy^2+dx+ey+f=0\;.$$ But that's no longer the equation of just a parabola: other conic sections like ellipses or hyperbolas fit that equation, too. For a parabola one has the discriminant $b^2-4ac=0$. In projective geometry this could also be seen as the line at infinity being a tangent to the conic section.

So one way to generalize from parabolas to degree three would start by observing that the generalization of a conic section (which is an algebraic curve of degree two) would be an algebraic curve of degree three, i.e. a cubic curve. Then among all these cubic curves, one would pick those with some special relationship to the line at infinity. One could pick the ones where the line at infinity is a tangent, with the the point of contact having algebraic multiplicity two. Or one could pick the ones which intersect the line at infinity in just a single point of multiplicity three. For the quadratic case, both these choices are the same, so either would be a valid generalization. Personally I'd prefer the former.

As the equation for a cubic curve has $10$ coefficients (instead of $6$ for a conic section), things become more complicated. For the cubic

$$ax^3+bx^2y+cxy^2+dy^3+ex^2+fxy+gy^2+hx+ky+l=0$$

the curve will have a tangent at infinity if and only if

$$b^2c^2-4ac^3-4b^3d+18abcd-27ad=0$$

since that's the discriminant here.

I don't know of any names or applications for such a class of cubic curves which are tangent to the line at infinity. Neither do I know of a geometric definition which would be an obvious generalization of the definition of a parabola by focus and directrix.