Cylindrical Wedge

$\begin{figure}\begin{center}\BoxedEPSF{CylindricalWedge.epsf}\end{center}\end{figure}$

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

$\begin{displaymath} hx-rz=0. \end{displaymath}$

(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

$\displaystyle x$	$\textstyle =$	$\displaystyle t$	(2)
$\displaystyle y$	$\textstyle =$	$\displaystyle \pm\sqrt{r^2-t^2}$	(3)
$\displaystyle z$	$\textstyle =$	$\displaystyle {ht\over r}$	(4)

for $t\in [0,r]$ . 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 ${\hbox{\sf R}}_x(90^\circ)$ and then by the Angle

$\begin{displaymath} \theta=\tan^{-1}\left({h\over r}\right) \end{displaymath}$

(5)

above the z-Axis. The transformed plane now rests in the

-plane and has parametric equations

$\displaystyle x$	$\textstyle =$	$\displaystyle {t\sqrt{h^2+r^2}\over r}$	(6)
$\displaystyle z$	$\textstyle =$	$\displaystyle \pm\sqrt{r^2-t^2}$	(7)

and is shown below.

$\begin{figure}\begin{center}\BoxedEPSF{CylindricalWedgeTongue.epsf scaled 800}\end{center}\end{figure}$

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

$\begin{displaymath} L=\sqrt{h^2+r^2} \end{displaymath}$

(8)

(and which follows immediately from the Pythagorean Theorem). The Volume of the wedge is given by

$\begin{displaymath} V={\textstyle{2\over 3}}r^2h. \end{displaymath}$

(9)

See also Conical Wedge, Cylindrical Segment