Conic sections are cross sections of a cone. They can be ellipses (including circles), hyperbolas, and parabolas. [br]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 [i]Connecting History to Secondary School Mathematics: An Investigation into Mathematical Intentions, Then and Now[/i], by Carrejo, Dennis, and Addington, to be published by Springer Verlag in 2025.[b]Manipulating the file[/b][br][list][*]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.[br][/*][/list]Warning: This file requires a lot of space to see and manipulate everything. Recommended: download it to a device with a big screen.