|
|
A cylindrical wedge, also called a cylindrical hoof or cylindrical ungula, is a wedge cut from a cylinder by slicing with a plane that intersects the base of the cylinder. The volume of a cylindrical wedge can be found by noting that the plane cutting the cylinder passes through the three points illustrated above (with ), so the three-point form of the plane gives the equation
(1)
| |||
(2)
|
Solving for gives
(3)
|
Here, the value of is given by
(4)
| |||
(5)
|
The volume is therefore given as an integral over rectangular areas along the x-axis,
(6)
|
Using the identities
(7)
| |||
(8)
| |||
(9)
| |||
(10)
|
gives the equivalent alternate forms
(11)
| |||
(12)
|
(Harris and Stocker 1998, p. 104). This simplifies in the case of to
(13)
|
The lateral surface area can be found from
(14)
|
where is simply with , so
(15)
| |||
(16)
| |||
(17)
| |||
(18)
|
(Harris and Stocker 1998, p. 104).
A special case of the cylindrical wedge which might be called a semicircular cylindrical wegde, is a wedge passing through a diameter of the base (so that ). Let the height of this 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
(19)
|
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
(20)
| |||
(21)
| |||
(22)
|
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
(23)
|
above the z-axis. The transformed plane now rests in the -plane and has parametric equations
(24)
| |||
(25)
|
and is shown below.
The length of the tongue (measured down its middle) is obtained by plugging into the above equation for , which becomes
(26)
|
(and which follows immediately from the Pythagorean theorem).
As determined from the case of the general cylindrical wedge, the volume of the semicircular cylindrical hoof is given by
(27)
|
and the lateral surface area by
(28)
|
The volume was found by Gregory of St. Vincent (1647).
While the centroid of the general cylindrical wedge is complicated for ,
(29)
|
for the semicircular cylindrical wedge, the centroid is given by
(30)
|
giving
(31)
| |||
(32)
| |||
(33)
|