# Plane Cubic Curves

Plane Cubic Curves

A conic section is a plane quadratic curve, that is, the graph of an equation of the form , for given coefficients . The possible curves are a circle, an ellipse, a parabola, or a hyperbola, with degenerate cases two straight lines (parallel or intersecting), a single straight line, a point, or the empty set.

a+bxy+c+dx+ey+f=0

2

x

2

y

a,b,…,f

A plane cubic curve is the graph of an equation of the form , for given coefficients . Cubics offer a far greater number of forms. There can be cusps, self intersections, an isolated point, a separate oval, and many combinations of straight, conic, or cubic asymptotes.

a+by+cx+d+e+fxy+g+hx+iy+j=0

3

x

2

x

2

y

3

y

2

x

2

y

a,b,…,j