Cylinder - surface contact and deformation

I’m trying to calculate the vertical movement (delta) of the center of a deformable cylinder in contact with non-deformable surface. The problem is presented on the picture in the link bellow. The problem is that I don’t really understand the point of the parameters z and w and I cannot find the equation for w.
z is supposed to be the vertical dimension of the point on the surface of the cylinder and x the horizontal. Their connection should be a parabolic function, which should describe the shape deformation of the cylinder next to the contact.

https://imgur.com/mmBN3zb

Does anyone have more insight into this? What am I missing here?