The cylindrical hoof is a special case of the cylindrical wedge given by a wedge passing through a diameter
of the base (so that ).
Let the height of the wedge be and the radius of the cylinder
from which it is cut be . Then plugging the points , , and into the 3-point equation for a plane
gives the equation for the plane as

(1)

Combining with the equation of the circle that describes the curved part remaining of the cylinder (and writing ) then gives the parametric
equations of the "tongue" of the wedge as

(2)

(3)

(4)

for .
To examine the form of the tongue, it needs to be rotated into a convenient plane.
This can be accomplished by first rotating the plane of the curve by about the x-axis
using the rotation matrix and then by the angle