Hyperbola - Maroon
Parabola - Red
Ellipse - Orange
Blue Dots - Foci
Green - Path of foci
This curve is a strophoid.
Thank you to Jean-Pierre Ehrmann of France for telling me this.

In sci.math Alexander Bogomolny wrote that this curve is traced by the point of intersection of the lines whose velocities are in the ration of 1:2. See Lighthouse
Clive Tooth followed Alexander's post
with a demonstration 
using a Dandelin sphere: 
Consider a cone.
Let the vertex of
the cone be V
and let g be a generator.
Call the axis of the cone x.
Let A be a fixed point on g. 
Let B be the point on x
which is closest to A.
Let r be the line through A
which is tangential
to the cone and which is
perpendicular to g.
(r is not visible
in this diagram because it's perpendicular to the screen)
Consider the pencil of
planes through r. 
Let P be one of these planes.
P will intersect our cone in some conic section. 
Consider a sphere, center C,
inscribed in the cone
and touching P, at a point F.
It is well known that F is a
focus of our conic section.
Clearly C is on x and the points
ABC and F are in the same plane.
the angle that AF
makes with g,
on the side of P
away from the sphere.
Call this angle 2t.
Now the angles
CBA and CFA are
both right angles.
So the points ABC & F
are concyclic.
So the angle ACF
is equal to
the angle ABF.
But the angle CAF is
(pi-2t)/2 = pi/2 - t,
so the angle ACF is t,
so the angle ABF is t.
Thus, A and B
are fixed points such that
for any position of
the plane P,
the angle gAF
is twice the angle ABF.
Thus, the point F
is the intersection
of two rotating lines
(BF and AF)
whose angular velocities
are in the ratio 1:2.

Thanks to Alexander and Clive, I can now see this curve as part of a moire pattern generated by two families of radial lines.