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Cylindrical Wedge


CylindricalWedgeSchem
CylindricalWedge

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 b>R), so the three-point form of the plane gives the equation

|x y z 1; R-b a 0 1; R-b -a 0 1; R 0 h 1|=h(R-b-x)+bz
(1)
=0.
(2)

Solving for z gives

 z=(h(x-R+b))/b.
(3)

Here, the value of a is given by

a=sqrt(R^2-(b-R)^2)
(4)
=sqrt(b(2R-b)).
(5)
CylindricalWedgeVolume

The volume is therefore given as an integral over rectangular areas along the x-axis,

 V=intz(x)y(x)dx 
=2int_(R-b)^R(h(x-R+b))/bsqrt(R^2-x^2)dx 
=h/(6b)[2sqrt((2R-b)b)(3R^2-2ab+b^2)-3piR^2(R-b)+6R^2(R-b)sin^(-1)((R-b)/R)].
(6)

Using the identities

a=Rsinphi
(7)
b=R(1-cosphi)
(8)
b^2=2bR-a^2
(9)
phi=1/2pi+tan^(-1)((b-R)/a)
(10)

gives the equivalent alternate forms

V=h/(3b)[a(3R^2-a^2)+3R^2(b-R)phi]
(11)
=(hR^2)/3((3sinphi-3phicosphi-sin^3phi)/(1-cosphi))
(12)

(Harris and Stocker 1998, p. 104). This simplifies in the case of a=b=R to

 V=2/3hR^2.
(13)

The lateral surface area can be found from

 S_L=2Rint_0^phiz(theta)dtheta
(14)

where z(theta) is simply z(x) with x=Rcostheta, so

S_L=2Rint_0^phi(h(b-R+Rcostheta))/bdtheta
(15)
=(2hR)/b{sqrt(2bR-b^2)+(b-R)×[1/2pi+tan^(-1)((b-R)/(sqrt(2bR-b^2)))]}
(16)
=(2hR)/b[a+(b-R)phi]
(17)
=2hR((sinphi-phicosphi)/(1-cosphi))
(18)

(Harris and Stocker 1998, p. 104).

CylindricalHoof

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 a=b=R). Let the height of this 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.
(19)

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
(20)
y=+/-sqrt(R^2-t^2)
(21)
z=(ht)/R
(22)

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)
(23)

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

x=(tsqrt(h^2+R^2))/R
(24)
z=+/-sqrt(R^2-t^2)
(25)

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)
(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

 V_S=2/3R^2h
(27)

and the lateral surface area by

 S_L=2Rh.
(28)

The volume was found by Gregory of St. Vincent (1647).

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,
(29)

for the semicircular cylindrical wedge, the centroid is given by

 x^_=int_0^Rint_(-sqrt(R^2-x^2))^(sqrt(R^2-x^2))int_0^(hx/R)xdzdydx,
(30)

giving

<x>=3/(16)piR
(31)
<y>=0
(32)
<z>=3/(32)pih.
(33)

See also

Cylindric Section, Cylindrical Hoof, Cylindrical Segment, Plücker's Conoid, Ungula, Wedge

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References

Gregory of St. Vincent. Opus Geometricum quadraturae circuli et sectionum coni. 1647.Harris, J. W. and Stocker, H. "Obliquely Cut Circular Cylinder" and "Segment of a Cylinder." §4.6.3-4.6.4 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 103-104, 1998.Kern, W. F. and Bland, J. R. "Truncated Prism (or Cylinder)." §31 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 81-83 and 127, 1948.

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Cylindrical Wedge

Cite this as:

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

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