The most well-known members of this family of two-dimensional curves are the circle and the ellipse. These arise when the intersection is a closed curve: the circle is a special case of the ellipse in which the plane is exactly perpendicular to the axis of the cone. If the plane is parallel to a generator line of the cone, the section is called a parabola. Finally, if the intersection is an open curve, and the plane is not parallel to a generator line of the cone, the figure is a hyperbola.
- If h2 = ab, this equation represents a parabola.
- If h2 < ab, this equation represents an ellipse.
- If h2 > ab, this equation represents a hyperbola.
- If a = b and h = 0, it represents a circle.
- If a + b = 0, it represents a rectangular hyperbola.
The eccentricity of a conic section is thus a measure of how far it deviates from being circular.
Conic sections are important in astronomy: the orbits of two massive objects that interact according to Newton's law of universal gravitation are conic sections if their common center of mass is considered to be at rest. If they are bound together, they will both trace out ellipses; if they are moving apart, they will both follow parabolas or hyperbolas.