The solid cut from a Cylinder by a tilted Plane passing through a Diameter of the base. It is also
called a Cylindrical Hoof. Let the height of the wedge be and the radius of the Cylinder from
which it is cut . 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 which 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 90° about the x-Axis using the Rotation Matrix
and then by the Angle
|
(5) |
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
|
(8) |
(and which follows immediately from the Pythagorean Theorem).
The Volume of the wedge is given by
|
(9) |
See also Conical Wedge, Cylindrical Segment
© 1996-9 Eric W. Weisstein
1999-05-25