Details

Consider a cylinder of radius with axis parallel to the axis. Its parametric equations are

,

,

,

where and are parameters.

Consider another cylinder, of radius with axis at a distance from the axis and where is the angle between the axis of the drill and the vertical. Its parametric equations are

,

,

,

where and are parameters.

The intersection curve of the two surfaces can be obtained by solving the system of three equations

for three of the four parameters .

Solving for , , and gives the parametric equations for the intersection curve with parameter :

,

,

The two parts of the equation represent the upper and lower half of the intersection curve.