How can we draw a Blaschke $3$ ellipse?

Question

Today I read the article Ellipses and Finite Blaschke Products by Ulrich Daepp, Pamela Gorkin, and Raymond Mortini.
In there they have proved very nice geometric results about per-images of Blaschke products which are fixed at zero of degree $2$ and $3$ on the unit circle.
Also in that paper they proved that the existence of marvelous geometric object called Blaschke $3$-ellipses which is looks like as in below.

Here I want to know about more geometrical properties of per-images of Blaschke products (similar in the above paper). Is there any?

Also I wonder how we can draw the pictures like Blaschke $3$ ellipses.
How we can find a software to draw a Blaschke $3$ ellipses?
or Can we draw them using a common software (like MATLAB)? How?

My question may be not a good one, but I am very curious about this. Thank you.

Answer 1

Here's how I'd plot your image in Mathematica.

B[a_, b_][z_] := z(z - a)/(1-Conjugate[a]z)(z-b)/(1-Conjugate[b] z);
compPts[a_, b_][lambda_] := 
   Quiet[Solve[B[a, b][z] == lambda, z], Solve::ratnz];
realPts[a_, b_][
   lambda_] := {Re[#], Im[#]} & /@ (z /. compPts[a, b][lambda]);
pic[a_, b_] := Graphics[{
    {Thick, Circle[]},
    {PointSize[Large], Point[{Re[#], Im[#]} & /@ {a, b}]},
    Line /@ 
     Table[Partition[realPts[a, b][Exp[I*t]], 2, 1, 1], {t, Pi/10, 
       2 Pi, Pi/10}]
    }];
pic[0.5, -0.4 + 0.4 I]

Now, the paper proves (as Theorem 1) that the envelope formed by those lines is exactly the ellipse with equation $$\left|z-a\right|+\left|z-b\right|=\left|1-\overline{a}{b}\right|.$$ I suppose the geometric properties that you refer to spring forth from that equation.