Cylindrical Hoof

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

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

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

above the z-axis. The transformed plane now rests in the -plane and has parametric equations

and is shown below.

The length of the tongue (measured down its middle) is obtained by plugging into the above equation for , which becomes

(and which follows immediately from the Pythagorean theorem).

As determined from the case of the general cylindrical wedge, the volume of the cylindrical hoof is given by

and the lateral surface area by

While the centroid of the general cylindrical wedge is complicated for ,

for the cylindrical hoof with , the centroid is given by

giving