Below is a simple applet showing the Poincaré model of the Hyperbolic plane.

To use the applet, you may simply click on two points in the disk and see the entire geodesic which contains them, or if you want a more precise way to locate points, you can input the x and y coordinates (or the real and imaginary parts if you'd prefer) of a point and it will be plotted. Also, as you move the mouse over the disk, you will see to the side the location of the mouse pointer.

If you right click inside the disk (alternately, hold shift and click, in case you don't have two mouse buttons), you'll pull up a menu. Thus far, this menu only has a few options: conjugate, reflect in a geodesic, remove a geodesic, rotate at a point, and apply a Möbius transformation. Conjugate does exactly what you would think; it applys complex conjugation to the whole picture. If you have clicked on a geodesic, then you may choose to reflect in that geodesic or remove it. If you choose to apply a Möbius transformation, then another window will pop up letting you input enough information to specify a Möbius transformation mapping the disk onto itself. In this window, you will be asked to specify an angle, and here you should give the angle in degrees since this is easier than asking for a multiple of pi. You will also be asked to input a complex number, which will be the number going to zero under this Möbius transformation. If this number's modulus is bigger than or equal to 1, nothing will happen. The format for rotating centered at a point is similar. The rotation will be centered at the point where you right-clicked.

Right now, the only thing that I know of that this does poorly is every once in a while it makes a bad decision about whether or not the points are joined with a diameter or a circular arc. What I think is happening is the circular arc that should be drawn has such a big radius and is so far out that there is enough roundoff error that the arc just doesn't quite look right (that's right; shockingly enough, when something goes bad I blame it on roundoff error). In this situation, I have seen the arc miss one or both of the points (but come pretty darn close to each). I've tried to find a happy medium so that it draws straight lines when they look best and arcs when they look best, but I won't claim it'll choose the right one every time.