Illustration of Conic Sections

The following four lines of code in Maple^TM produce this animation of the intersection of a plane of increasing slope with a right circular cone.



> with(plots):
> C := plot3d([z*cos(t), z*sin(t), z], t = -Pi..Pi, z = -3..5, transparency = .6):
> P := animate(plot3d, [1+x*tan(t), x = -3..5, y = -3..5], t = 0..1.57, frames=51, view = -3..5, orientation = [-135, 35]):
> display([C, P]);

(t is the angle in radians between the moving plane and the xy plane)

When the plane is parallel to the xy plane (t =0, perpendicular to the axis of symmetry of the cone), the intersection is a circle (eccentricity e = 0) .
As the slope of the plane increases, the intersection passes through ellipses of increasing eccentricity 0 < e < 1 , until,
half way through the animation, (t = p/4) , the plane is parallel to one edge of the cone.   The intersection is then a parabola e = 1 .
Thereafter, e > 1 , the steeper plane also intersects the lower half of the cone, producing the two branches of the hyperbola.
Finally, when the plane is vertical (parallel to the axis of the cone), the plane passes through the origin, e is infinite and the conic section becomes a line pair.

In Maple one can control the playback speed or step frame by frame.
One can also adjust the viewing angle and other parameters within Maple.

Maple also has a very nice demonstration of conic sections on its extensive help site.
From the menu bar in Maple, select   Tools   then   Math Apps.
In the new window, click on   Algebra and Geometry   then   Conic Sections.

Maple Worksheet   conics.mw

Created 2018 09 07 and most recently modified 2022 08 26 by Dr. G.H. George