Conics

A conical surface is formed by the rotation of a line, g, (also called the generatrix or generator) around another line, e, the axis, which is fixed at the point V, the vertex or apex, which forms the angle α.

Conics

The nappes are the upper and lower portions of the conical surface which is divided by the appex.

A conic section is the area of intersection of a conical surface with a plane that does not pass through its apex. Depending on the relationship between the angle of a conical surface, α, and the inclination of the plane with respect to the axis, β, the conic sections will be elipses, circles, parabolas or hyberbolas.

Ellipse

Ellipse as a Conical Surface


α < β < 90º

If the plane cuts all the generators (of one nappe only) and it is not perpendicular to the axis, the conic section is an ellipse.

Circle

Circle as a Conical Surface


If β = 90°, that is to say, the plane is perpendicular to the axis, the conic section is a circle.

Parabola

Parabola as a Conical Surface



α = β

If the plane intersecting the conical surface is parallel to the generatrix, the conic section is a parabola.

Hyperbola

Hyperbola as a Conical Surface



α > β

If the plane intersects two nappes of the conical surface and the angle of the generatrix, α, is greater than the inclination of the plane with respect to the axis, β, the conic section is a hyperbola.