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Cylindrical Hoof


CylindricalHoof

The cylindrical hoof is a special case of the cylindrical wedge given by a wedge passing through a diameter of the base (so that a=b=R). Let the height of the wedge be h and the radius of the cylinder from which it is cut be 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

 hx-Rz=0.
(1)

Combining with the equation of the circle that describes the curved part remaining of the cylinder (and writing t=x) then gives the parametric equations of the "tongue" of the wedge as

x=t
(2)
y=+/-sqrt(R^2-t^2)
(3)
z=(ht)/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 degrees about the x-axis using the rotation matrix R_x(90 degrees) and then by the angle

 theta=tan^(-1)(h/R)
(5)

above the z-axis. The transformed plane now rests in the xz-plane and has parametric equations

x=(tsqrt(h^2+R^2))/R
(6)
z=+/-sqrt(R^2-t^2)
(7)

and is shown below.

CylindricalWedgeTongue

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

 L=sqrt(h^2+R^2)
(8)

(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

 V=2/3R^2h
(9)

and the lateral surface area by

 S_L=2Rh.
(10)

While the centroid of the general cylindrical wedge is complicated for R!=b,

 x^_=int_(R-b)^Rint_(-sqrt(R^2-x^2))^(sqrt(R^2-x^2))int_0^(h(b-R+x)/b)xdzdydx,
(11)

for the cylindrical hoof with R=a=b, the centroid is given by

 x^_=int_0^Rint_(-sqrt(R^2-x^2))^(sqrt(R^2-x^2))int_0^(hx/R)xdzdydx,
(12)

giving

<x>=3/(16)piR
(13)
<y>=0
(14)
<z>=3/(32)pih.
(15)

See also

Cylindrical Wedge

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Cite this as:

Weisstein, Eric W. "Cylindrical Hoof." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CylindricalHoof.html

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