# Explicit parametrization for a 3-ellipse? A 4-ellipse?

I searched around but was unable to find anything.

For the usual $$2$$-ellipse we have the parametrization $$x(t) = a\cos(t)$$ and $$y(t) = b\sin(t)$$ for $$t\in [0,2\pi]$$.

Is there anything similar for the more general $$3$$-ellipse or $$4$$-ellipse? If there general case is too difficult is there a parametrization for a constrained version, for example where some of the foci lie, say on the $$x$$-axis?

Edit: I mean in 2 dimensions, as here https://en.wikipedia.org/wiki/N-ellipse

Edit 2: Assume $$u_1 = (-R,0)$$ and $$u_2 = (R,0)$$ and $$u_3 = (0, -H)$$ are the foci of the $$3$$-ellipse, for $$R,H > 0$$, given some $$d$$ the $$3$$-ellipse is the set of points given by $$\left\{(x, y) \in \mathbf{R}^{2} : \sum_{i=1}^{3} \sqrt{\left(x-u_{i}\right)^{2}+\left(y-v_{i}\right)^{2}}=d\right\}$$

Is there a parametrization of this curve similar to the $$2$$-ellipse?

• I had never heard of these multifocal ellipses, and find your question very interesting. I see that even a trifocal ellipse will be an algebraic curve of degree eight, which suggests to me that any parametrization will be extremely difficult to find. What if the foci are the vertices of an equilateral triangle? Mar 15, 2019 at 3:05
• To avoid a confusion of the number of the foci with the number of dimensions, add a definition of n-ellipse and optionally, some examples (pictures) to the question. Mar 15, 2019 at 3:11
• I will edit the question when I get back home, stuck with mobile for now Mar 15, 2019 at 10:45