Plane Cubic Curves

plot range

5

10

20

coefficients

3

x

1.

2

x

y

0

x

2

y

0

3

y

-1.

2

x

0

x y

0

2

y

0

x

0

y

0

1

10

A conic section is a plane quadratic curve, that is, the graph of an equation of the form

a

2

x

+bxy+c

2

y

+dx+ey+f=0

, for given coefficients

a,b,…,f

. 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 plane cubic curve is the graph of an equation of the form

a

3

x

+b

2

x

y+cx

2

y

+d

3

y

+e

2

x

+fxy+g

2

y

+hx+iy+j=0

, for given coefficients

a,b,…,j

. 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.