1. The
implicit equation in
the xyplane is
where a and b are positive constants. This is a
curve
with reflectional symmetry about both the horizontal and vertical axes.
2. Any Hippopede is the intersection of a torus
with one of its
tangent planes that is parallel to its axis of rotational symmetry, as
demonstrated in the animation.
3. Converting the implicit equation to polar
coordinates gives
so the origin at r = 0 is a solution, and the
remainder
of the curve is given by
.
4. If 0 < b< a,
the point
at the origin is an isolated node and the balance of the solution set
is
a simple closed curve, also called an Elliptic Lemniscate
of
Booth.
The b < a special case is
given by the rational parametrization
Each of these curves is the image of an ellipse under an
inversion in
a circle. The circle and the ellipse must have the same
center.
The curve is nonconvex for b < a
<
2b, and convex for a > 2b
when the
shape is oval, but is not exactly a strictly defined ellipse.
5. If a = b, the
quartic implicit equation
factors into two quadratics; thus, the curve is the union of two
circles, centered at (  b,0
)
and ( b, 0 ), each of radius b, and mutually
tangent
at the origin.
6. If 0 < a < b,
the
curve intersects itself at the origin, and it is also called a Hyperbolic
Lemniscate of Booth.This is given by the rational
parametrization
Each
of these curves is the image of a hyperbola under an inversion in a
circle. The circle and the hyperbola must have the same
center.

The animation.
Now that you have
investigated the equations and read of the special properites, we
suggest you use the "Reload" or "Refresh" button at the top of your
computer. This will replay the animation. Visually "slice"
the torus to see the plane lemniscate evolve into a pair of circles,
and then the oval shape.
and
