The figure below displays the graph of the distance function y=d(A,B)=f(x) of two parallel tangents at points X and B of the ellipse c and circle c', the center of c' lying on the minor axis of the ellipse. Function f(x) is a periodic one depending on x which measures the polar angle of X (x is the so-called eccentric angle of the ellipse). The figure contains only the graph over a period of the function i.e. the graph over the interval [0, 2*pi].
Depending on the relative position of the two circles and the shape of the ellipse, function f(x) may have four or two local extrema inside its period-interval. In the particular case in which c is a circle concentric to c' the function is of course a constant. The magnifying glass shows the behaviour of the function (magnified by a factor of 50) in the neighborhood of a local minimum M.
![[alogo]](alogo.png){kind=small}
A distance function ![[0_0]](DistanceFunction_files/DistanceFunction_0_0_0.png)
![[0_1]](DistanceFunction_files/DistanceFunction_0_0_1.png)
![[0_2]](DistanceFunction_files/DistanceFunction_0_0_2.png)
![[1_0]](DistanceFunction_files/DistanceFunction_0_1_0.png)
![[1_1]](DistanceFunction_files/DistanceFunction_0_1_1.png)
![[1_2]](DistanceFunction_files/DistanceFunction_0_1_2.png)