Parabolic cross 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 the plane.To get a parabola, the plane must be parallel to a tangent plane to the cone. To construct this, first you will need a plane parallel to the base of the cone, giving a circular cross section (shown here at A). The tangent line to the circle at A is perpendicular to the radius at A. Then the tangent plane to the cone at A contains the tangent line to the circle and the line through A and the vertex of the parabola, AO. Translate the plane off the tangent position to get the plane you want.The proof that this cross section is a parabola is too long and complicated to give here. It 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 parabola 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.
- To see what happens for different positions, move the point A around the circle.
- Use the slider to change the distance of the parabola plane from the tangent plane.
- Change the shape of the cone with the sliders ShearFactor and VerticalStretch.