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)
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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)
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(3)
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(4)
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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
(5)
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above the z-axis. The transformed plane now rests in the -plane
and has parametric equations
(6)
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(7)
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and is shown below.
The length of the tongue (measured down its middle) is obtained by plugging into the above equation for
, which becomes
(8)
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(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
(9)
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and the lateral surface area by
(10)
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While the centroid of the general cylindrical wedge is complicated for ,
(11)
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for the cylindrical hoof with , the centroid is given by
(12)
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giving
(13)
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(14)
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(15)
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