This interactive diagram allows you to manipulate a plane to get any conic section from a cone.
It's hard to describe the position of a plane. In this diagram, you can change the plane in three ways:
The "Change tangent line direction" point changes the blue line on the cone through the vertex. This line determines a plane tangent to the cone, but the plane is not shown (to reduce clutter.)
The "Rotate section plane" point moves on the green arc, and changes the direction of the plane shown. At the ends of the arc, the section plane is parallel to the (invisible) tangent plane.
The "Move section plane" point moves the plane up or down without changing its direction. (For strange technical reasons, this point moves horizontally, causing the plane to move vertically.)
See if you can get all three kinds of conic section: ellipse, hyperbola, parabola.
How is the direction of the plane (such as parallel to a tangent to the cone) related to the type of curve you get?
