construct an optimal car ramp minimizing the possible ride height

Question

In the words of mathematics there is the problem of connecting two horizontal street sections separated by a distance, say A, at different height levels (difference H) with a function f(x). You now can imagine a secant line of given length "traveling" along the ramp while the maximal distance D from secant line to the function f(x) is measured. The objective is to find a function f(x) which minimizes this distance D for all secant lines. Returning to the initial problem, the shape of this function f(x) would allow a maximum of cars to use this ramp unharmed. I am interested whether any result can be obtained by calculus of variations or a solution has to be found numerically in any special case.