The upper half-plane model of hyperbolic $2$-space is $$\mathcal{H}^2:=\mathbb{R}\times\mathbb{R}^+$$ with metric $$ds=\sqrt{\frac{dx^2+dy^2}{y^2}}.$$ Following are some interesting features of this space. Straight lines according to this metric are vertical half-lines and half-circles perpendicular to $\mathbb{R}$. Points on $\mathbb{R}$ do not lie in the space, but rather lie infinitely far from any point in the space, in the sense that the distance from any fixed point to some other point $p$ gets arbitrarily large as $p$ gets arbitrarily close to $\mathbb{R}$. A circle with a center in $\mathbb{H}^2$ looks like a Euclidean circle, however the center looks (in the Euclidean sense) like it is closer to the bottom of the circle.
I'm interested in seeing parametrized curves in this space. I'd especially love if there is a user-friendly graphics program that will graph curves according to the hyperbolic metric. What I'm talking about is a bit different from just an analogy of graphing functions of one variable in $\mathbb{R}^2$, because it would look the same. Instead I'm talking about functions that are expressed in terms of hyperbolic distances. So the first part of the question (rather vaguely) is whether there is some good software available to do that.
I'm especially interested in the following type of curve. Fix two points $a,b\in\mathcal{H}^2$, fix a constant $K\in\mathbb{R}$, and define $$S_{a,b,K}:=\big\{p\in\mathcal{H}^2\mid\mathrm{dist}_{\mathcal{H}^2}(a,p)^2-\mathrm{dist}_{\mathcal{H}^2}(b,p)^2=K\big\}.$$ What does $S_{a,b,K}$ look like? How does it change as we vary $K$?
Here is a simplified version. Suppose $p$ and $q$ are the same height (same second coordinate). If $K=0$, then $S_{a,b,K}$ is a vertical line passing between $p$ and $q$. What happens to $S_{a,b,K}$ in this case as $K$ moves away from $0$? It does not remain a vertical line. It also does not become a half-circle (the other type of hyperbolic line). But it is some kind of curve. Ideally I'd like to play around with some graphics software that allows me to see this.