A hyperbolic lens has extremely good focusing qualities. Using the same central curvature and thickness that were used for the spherical and parabolic lenses I was able to get good results by adjusting one free constant. The plots are given in centimeters.
Zooming in on the focal point one can see that the dimensions of the focal region is measurable in micrometers (μm = 10-6 meters).
The hyperbola's formula, x(y), assumes that the it passes through the origin so one needs to add or subtract a constant to shift it slightly left or right. I was able to eliminate the scale factor, a, for the x axis by solving for a value which gives the curvature, κ0, at the origin of the function. The remaining scale factor, b, for the y axis was then available to adjust the focusing properties of the lens.