For a nonzero level , completing the square gives
(1)
Thus the level sets of a linear numerator over a radial quadratic are circles centred on , real only for . The extreme levels shrink to points. The zero contour is the straight line , separating positive and negative heights; near it, small nonzero levels are large circles. Heights tend to zero at infinity.
Figure 1.
Circular nonzero contours around the two extrema of a rational height function
. Original contour plot. Blue positive and orange negative circles surround the marked maximum and minimum; the black line is the zero contour. Circle centres lie on . Details.

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