I am a little confused. You are asking for solutions to $p^2 + q^2 = d^2,$ if I am not mistaken. If the solutions are to be relatively prime, there is a well-known parametrization of the pairs $(p, q),$ namely $(2u v, u^2-v^2)$, and of course the symmetric set $(u^2-v^2, 2 u v)$ (and the images flipping $u$ and $v.$). The non-relatively-prime solutions are multiples of these. So, presumably the curves you are seeing are rational curves coming from this, and they foliate the entire solution spaces.

I am a little confused. You are asking for solutions to $p^2 + q^2 = d^2,$ if I am not mistaken. If the solutions are to be relatively prime, there is a well-known parametrization of the pairs $(p, q),$ namely $(2u v, u^2-v^2)$, and of course the symmetric set $(u^2-v^2, 2 u v)$ (and the images flipping $u$ and $v.$). The non-relatively-prime solutions are multiples of these. So, presumably the curves you are seeing are rational curves coming from this, and they fill the entire solution space.