Addition from notes on July 9, 2017.

Define radius function:

$$r(\phi) =\sqrt{ 2 \csc(2\phi)^2\bigg(1-\sqrt{1-\alpha \sin(2\phi)^2}\bigg) },$$

The solution of

$$ \alpha = x^2 + y^2 - x^2 y^2 , $$

Then compute, as above,

$$\frac{dt}{d\phi} = \frac{d}{d\alpha} r(\phi,\alpha)^2= \frac{1}{\sqrt{1-\alpha \sin^2(2\phi)}},$$

thus completing the Hamiltonian Analogy. This provides an alternative, geometric answer to the question: how should the addition law be analyzed? ( cf. HALES ). More on this later...

As for now, how about another picture? Generated from a slight modification of the algorithm above, negative $\alpha$ solutions along the Edward's curve:

Available on Shapeways soon ! !