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[url=http://plane.to/]plane. To[/url]get an ellipse, the plane must cross only one of the nappes (upper or lower section) of the cone. [url=http://plane.to/]To[/url]get an ellipse, the plane must cross both of the nappes of the cone. [br][br]An oblique cone has [i]two[/i] 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. [br][br]See the GeoGebra applet [url=https://www.geogebra.org/m/axj59hzd]"Parabolic cross sections of a cone"[/url] for a construction of parabolas. [br][br]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 [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.[br][br][b]Manipulating the file[/b][list][*]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.[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.