Cut the Cone

Conic sections are the curves that may be obtained by cutting cones with planes. The applet below is intended to assist you in doing just that.

Roughly speaking, the applet displays two objects: a 2-nappe cone and a plane that can be controlled with a mouse separately and also simultaneously depending on which of the "radio buttons" at the bottom of the applet is checked.

Objects can be rotated with the left button pressed or translated with the right button pressed. Pressing the left button while the Alt key is down lets one zoom in and away from an object. When the "combined" button is checked the cone will move as usual but also together with the plane.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

What if applet does not run?

The cone as well as the plane should of course be assumed with no bounds. The cone has an axis of symmetry passing through the vertex; the plane perpendicular to the axis cuts off a circle which you see in the applet since the two objects shown are both finite. The shape of the conic section cut off by a plane depends on two parameters:

  1. angle α formed by any of the cone generators (the line on the surface of the cone passing through the vertex) and the base,
  2. angle β between the base and the cutting plane.

Both angles are assumed acute. In addition, β my be zero in which case the cutting plane is perpendicular to the axis or, which is the same, parallel to the base. The possibilities are as follows:

  1. β < α: the plane cuts the cone in an ellipse or, if it passes through the vertex, a point,
  2. β = α: the plane cuts the cone in a parabola or, if it passes through the vertex, a line.
  3. β > α: the plane cuts the cone in a hyperbola or, if it passes through the vertex, a pair of lines.

In #1, if β = 0, the ellipse becomes a circle.