Mark each point of $\mathbb{N}^2$ ($\mathbb{N}$ the natural numbers) if its
Euclidean distance from the origin is an integer. One obtains a plot like this, symmetric about the $45^\circ$ diagonal.

![IntDists0](https://i.stack.imgur.com/4J3MB.jpg)  

There are obvious patterns here. The straight lines through the origin are derived
from scalings of [Pythagorean triples](http://en.wikipedia.org/wiki/Pythagorean_triple): the $(3,4,5)$ triangle, the $(5,12,13)$ triangle, the $(8,15,17)$ triangle, etc.
But other patterns are discernible, some of which I am perhaps hallucinating:

![IntDists1](https://i.stack.imgur.com/5wfYp.jpg)  

Do these patterns reflect Diophantine curves dense in integer-distance points?
To what extent is, in some sense, this entire plot (extended indefinitely) understood
as a union of such curves?
Or do there remain unknowns lurking in here, i.e., there are sporadic
points with no as-yet apparent logic behind their appearance?

(Tangentially related to the MO question, "[Integer-distance sets](https://mathoverflow.net/q/136925/6094)".)

---

**Addendum**. It is now clear (after looking at plots extending further)
that the blue and tan curves are actually one, crossing the diagonal,
and not pinched off as I drew them.