Geometric Origin of the Conic Sections

In this section, we are interested in the classical construction of the conic sections. Note the following diagram: INSERT DIAGRAM OF PLANE CUTTING RIGHT--CIRCULAR CONE

It was Apollonius (mentioned above) who was the first to note that the conic sections could be constructed apart from algebraic equations by cutting the right--circular cone with a plane. As a matter of fact, Apollonius did not note the connection of the conics to their algebraic equations. These equations did not enter the mathematical picture for hundreds of years.

Let us examine the figure above a little more closely. Note that if the right--circular cone is cut by a plane which is perpendicular to the axis of the cone, then the resulting intersection is a circle. (Note that the axis of the cone in the figure is simply the vertical line which goes through the vertex of the cone, the point at which the two pieces of the cone meet.) If the cutting plane is parallel to the ``side'' of one of the cones, then we see that a parabola is constructed. Thirdly, if the plane intersects both pieces of the cone, then a hyperbola appears. Finally, if the plane intersects one of the pieces of the cone but is NOT parallel to the ``side'' of one of the cones, then an ellipse is generated.