The above model allows for both the Dens and the C2 spinous to be displaced from the central ray. The amount of C2 rotation wrt the central ray can be calculated by plugging in reasonable values for the distances influenced by x-ray placement.
I generated a spreadsheet from the equation derived above and used it to calculate a family of curves to show how a measured spinous deviation would relate to C2 rotation in the coronal plane. The Tube-to-Film Distance (TFD) is fixed at 40 inches. The Dens-to-film distance (DFD) is 7.5 inches, and the spinous to film distance (SFD) is 6.0 inches.
You can see that the top curve represents the case when the Dens is dead on the central ray. In this case C2 rotation in degrees is a nearly linear function of C2 spinous deviation from the midline. You would expect to see 0.7 inches of spinous deviation for 20 degrees of rotation.
You can also see how similar the curves are in the range shown, being only offset in the y-direction with changing values of dens off-centering. So, it doesn't matter much where the dens is wrt the Central Ray, you could just as well assume the dens IS dead-on and just measure the distance from the center of the dens to the spinous of C2. For example, the following table shows the amount of error you might get by just measuring the distance from the dens to the spinous.
For instance, when the dens is offset by 0.25 ", and you ignore the offset, then the error introduced is only 0.4 degrees. Not a problem. The error only becomes significant if the dens is offset by about a half inch.
Question: How does this compare to the rotation calculation used in the Grostic Procedure?