Joint distribution of sphere, cylinder, and axis

A mathematical expression for the joint probability density function of sphere, cylinder, and axis is presented for the first time. It holds under certain circumstances. A relation is derived which may be useful in obtaining the joint probability function when that expression does not hold. The joint probability density function is of fundamental importance in all quantitative studies of dioptric power in the form sphere, cylinder, and axis. Some of its properties are described. It represents a hypersurface in a four-dimensional space with a maximum, a saddle point, and variety of ridges.