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Cylindrical Wedge


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 $h$ and the radius of the Cylinder from which it is cut $r$. Then plugging the points $(0, -r, 0)$, $(0, r, 0)$, and $(r, 0, h)$ into the 3-point equation for a Plane gives the equation for the plane as

\end{displaymath} (1)

Combining with the equation of the Circle which describes the curved part remaining of the cylinder (and writing $t=x)$ 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
\theta=\tan^{-1}\left({h\over r}\right)
\end{displaymath} (5)

above the z-Axis. The transformed plane now rests in the $xz$-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 $t=r$ into the above equation for $x$, which becomes

\end{displaymath} (8)

(and which follows immediately from the Pythagorean Theorem). The Volume of the wedge is given by
V={\textstyle{2\over 3}}r^2h.
\end{displaymath} (9)

See also Conical Wedge, Cylindrical Segment

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© 1996-9 Eric W. Weisstein