# Conical Surface Juggling

Xah Lee, 2006-11, 2010-11-02

Greg Kennedy juggling inside a conical enclosure.

It looks like juggling in wonderland, doesn't it?

You may not know, that when a cone is cut by a plane, their intersection forms a curve on the cutting plane, and this curve is mathematically classified into 3 types: ellipse (closed), hyperbola (open), parabola (open), depending the angle of the cutting plane with respect to the cone.

In general, these curves are literally and collectively called Conic Sections, or just Conics, and is the first systematically studied math subject by ancient Greeks.

You can see illustrations and their math details here: conic sections, ellipse, hyperbola, parabola.

Now, this is why, when the guy throws the ball even towards the ground, the ball comes up higher.

As long as he does not aim the ball precisely at the tip of the cone (his feet), then the ball will make a dip and come up!

Also of interest, is that the intersection of a cylinder and a plane is also a ellipse. This means, if the glass wall is a cylinder, then a ball thrown in the downward direction will also come up.

In practice, gravity will make some of these impractical. And the cone can't be perfect with a pinnacle because else he wouldn't have a place to stand on.

## Juggling on Surface

Greg Kennedy's conics juggling is interesting because it force the balls on a surface. Normal juggling, is a 3D affair. By making balls roll on a surface, the juggling becomes a 2D affair, and is in fact easier for the same number of balls.

However, note that in practice, most juggling of more than 3 balls are essentially planar. For example, imaging juggling 5 balls. Although there are no constraints on where the ball would fly, but in practice it is thrown such that its path lies on the plane parallel to the juggler's face.

Greg's juggling is interesting because by forcing the balls to roll on a surface, it visually accentuate one beautiful aspect of juggling: the mathematical patterns made by the ball's paths.

### Surface Juggling on a Hemisphere

There are many of mathematical surfaces. We have the plane, cylinder, cone. These are all rather “flat”. They are part of what's called ruled surface, because they can all be generated by a single line. Then, there's sphere, torus, and many other weird surfaces. So, what happens if you juggle on a sphere?

Surface Juggling on a Hemisphere

## Bounce Juggling on Two Orthogonal Planes

Bounce Juggling on Two Orthogonal Planes