h-length parallel curve

The h-length is measured, relative to O, along the segments B_nA_n, which are straight lines parallel to the x axis. When the h-length is a constant, the end points of the segments form an h-length parallel curve.

In the diagram, the h-length of any segment B_nA_n, relative to O, is 2.

The equation for the h-length parallel curve is:

x = y(e^2(a)/y - 1)^½
a equals the h-length constant

in the diagram:
the value of y at B₃ is 6
a = 2

the value of x at A₃ is

x = 6(e^2(2)/6 - 1)^½
= 6(1.947734 - 1)^½
= 5.84109