One of the earlier descriptions of the shape of this curve was given by Hill (1910).

From the equilibrium equation: Hb_n + n O₂ <=> (HbO₂)_n

Came the mathematical description:  Saturation = K* PO₂ⁿ/(1 + KPO₂ⁿ)

With a value for n of around  2.5

It was determined that Haemoglobin consisted of 4 subunits.  Adair (1925) postulated 4 equilibrium reactions with constants generating an equation of the form:

Saturation = (a₁p+2a₂p2+3a₃p3+4a₄p4)/[4*(a₁p+a₂p2+a₃p3+a₄p4)]

where a₁-a₄ are the "Adair parameters" and p is the oxygen tension.

This approach is appealing because it seems to reflect the underlying mechanics of cooperative oxygen binding.  Sadly it is an oversimplification of the process.  However it remains a basis of many mathematical models. (Kelman 1966)

This program uses the equation of Severinghaus (1979) as it seems to have a nice balance of accuracy and simplicity.

 Saturation = [23,400*(PO₂³+150*PO₂)^-1+1]^-1

An inversion of this equation to determine Oxygen tension from saturation has also been described although the mathematics is not for the faint hearted. (It contains the cube root of a negative number!) (Ellis 1989)

Adair, G. S. (1925). “The hemoglobin system. VI. The oxygen dissociation curve of hemoglobin.” J Biol Chen 63: 529-545.

Ellis, R. K. (1989). “Determination of PO2 from saturation [letter].” J Appl Physiol 67(2): 902.

Hill, A. V. (1910). “The possible effects of the aggregation of the molecules of haemoglobin on its oxygen dissociation curve.” J Physiol (Lond) 40: 4-7.

Kelman, G. R. (1966). “Calculation of certain indices of cardio-pulmonary function, using a digital computer.” Respir Physiol 1(3): 335-43.

Roughton, F. J. and J. W. Severinghaus (1973). “Accurate determination of O2 dissociation curve of human blood above 98.7 percent saturation with data on O2 solubility in unmodified human blood from 0 degrees to 37 degrees C.” J Appl Physiol 35(6): 861-9.