How many points can a circle in space have in common with a Cassini oval in 3D? Given :
A circle of radius $R$ with its center at the $XYZ$-origin (in space).
Two different fixed points in space $(x_{1},y_{1},z_{1})$ and $(x_{2},y_{2},z_{2})$ that can lie anywhere.
The product of the distances to these points equals a given constant $C$.
The latter two conditions define a Cassini oval.
How many points can this circle in space have in common with a Cassini oval in 3D?
This geometric puzzle creeps up in a project where I want to find out when/where signal strength will be maximum in a bistatic setup with a fixed transmitter and receiver, and a moving aircraft.
 A: The intersection of the Cassini oval with the plane holding the circle is a quartic curve. By Bézout's theorem, when the number of intersection of that quartic curve with the circle is finite, then it is at most $8 = 4\times 2$. 
Please note that it is possible for the quartic curve to intersect the circle at infinite many places. This can happen if the circle and Cassini oval share a common symmetry axis.
Update
It turns out for Cassini oval, when the number of intersection is finite, the maximum number of intersection is $4$ instead of $8$.
WOLOG, choose a coordinate system such that the circle lies on the $z = 0$ plane
with equation $$\mathcal{C} : x^2 + y^2 = r^2$$
If one "square" the defining relation of the Cassini oval and intersect it with
the $z = 0$ plane, its equation will have the form
$$\mathcal{O} : (x^2+y^2 + L_1(x,y))(x^2+y^2+L_2(x,y)) = C^2$$
where $L_i(x,y) = a_i x + b_i y + c$ are affine in $x,y$.
Notice for any $(x,y) \in\mathcal{C}$,
$$(x,y) \in \mathcal{O}\quad\iff\quad (r^2 + L_1(x,y))(r^2+L_2(x,y)) = C^2$$
The last equation defines a conic $\mathcal{E}$ and $\mathcal{C} \cap \mathcal{O} = \mathcal{C} \cap \mathcal{E}$.
By Bézout's theorem again, when the intersection is finite, the number of intersection is at most $4 = 2 \times 2$.
A: The section of the hyperboloid by the plane of support of the circle is a conic. Two conics can intersect in four points.
A: Deriving the implicit equation for the 3D Cassinian is easy, $2a$ is the distance between the fixed points, $b$ is the constant.
When the foci lie symmetric to the $XYZ$ origin on the $Y$ axis, the surface can be neatly visualised in Wolfram Mathematica, using :
ContourPlot3D[(x^2+(y+a)^2+z^2)*(x^2+(y-a)^2+z^2)==b^2,{x,-2*a,2*a},{y,-2*a,2*a},{z,-2*a,2*a},WorkingPrecision->16,PlotRange->Automatic,BoxRatios->Automatic,PlotPoints->50,ContourStyle->Directive[Cyan,Opacity[0.5]],Mesh->None,AxesLabel->{"X","Y","Z"},Ticks->None]

The $a$ is the known euclidean distance between transmitter - receiver, the constant $b$ can be calculated as well, so I'll be able to see what shape the particular Cassinian takes and how the circle (aircraft's track) intersects it.
