Elliptic and hyperbolic sections of a cone
Conic sections are cross sections of a cone. They can be ellipses (including circles), hyperbolas, and parabolas.
To get the complete curves, the cone must be an infinite cone, in two directions. The cone does not have to be a right cone (one whose vertex is directly above the center of its base circle). If you pick a random plane for the cross sections, almost all of the time you will get a non-circular ellipse or a hyperbola. To get a circle or a parabola you must be very specific about theplane. Toget an ellipse, the plane must cross only one of the nappes (upper or lower section) of the cone. Toget an ellipse, the plane must cross both of the nappes of the cone.
An oblique cone has two directions of planes that give circular sections. One of them is the base circle of the cone (horizontal). For the other, move the point A to H or the point opposite H on the circle, then adjust the angle.
See the GeoGebra applet "Parabolic cross sections of a cone" for a construction of parabolas.
The proof that these cross sections are ellipses and hyperbolas is too long and complicated to give here. An outline of the proof is in the book Connecting History to Secondary School Mathematics: An Investigation into Mathematical Intentions, Then and Now, by Carrejo, Dennis, and Addington, to be published by Springer Verlag in 2025.
Manipulating the file
- The window at right shows the curve in its own plane.
- Use the Rotate 3D Graphics View tool or other 3D Graphics View tools to look at the objects from a different viewpoint.
- Use the Angle slider to change the angle of the section plane.
- To see what happens for different positions, move the point A around the circle.
- Change the shape of the cone with the sliders ShearFactor and VerticalStretch.