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Plücker's Conoid


PluckersConoid

Plücker's conoid is a ruled surface sometimes also called the cylindroid, conical wedge, or conocuneus of Wallis. von Seggern (1993, p. 288) gives the general functional form as

 ax^2+by^2-zx^2-zy^2=0,
(1)

whereas Fischer (1986) and Gray (1997) give

 z=(2xy)/((x^2+y^2)).
(2)

A polar parameterization therefore gives

x(r,theta)=rcostheta
(3)
y(r,theta)=rsintheta
(4)
z(r,theta)=2costhetasintheta.
(5)

The cylindroid is the inversion of the cross-cap (Pinkall 1986).

Plucker2Plucker3Plucker4

A generalization of Plücker's conoid to n folds is given by

x(r,theta)=rcostheta
(6)
y(r,theta)=rsintheta
(7)
z(r,theta)=csin(ntheta)
(8)

(Gray 1997), which is a slight variation of the form called the "conical wedge" by von Seggern (1993, p. 302).

The coefficients of the first fundamental form of the generalized Plücker's conoid are

E=1
(9)
F=0
(10)
G=1/2{2r^2+c^2n^2[1+cos(2nt)]}
(11)

and of the second fundamental form are

e=0
(12)
f=-(sqrt(2)cncos(nt))/(sqrt(2r^2+c^2n^2[1+cos(2nt)]))
(13)
g=-(sqrt(2)cn^2rsin(nt))/(sqrt(2r^2+c^2n^2[1+cos(2nt)])).
(14)

The Gaussian and mean curvatures are given by

K=-(4c^2n^2cos^2(nt))/({2r^2+c^2n^2[1+cos(2nt)]}^(3/2))
(15)
H=(sqrt(2)cn^2rsin(nt))/({2r^2+c^2n^2[1+cos(2nt)]}^(3/2)).
(16)

See also

Cross-Cap, Cylindrical Wedge, Right Conoid, Ruled Surface, Wedge

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References

Fischer, G. (Ed.). Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Kommentarband. Braunschweig, Germany: Vieweg, pp. 4-5, 1986.Gray, A. "Plücker's Conoid." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 435-437, 1997.Pinkall, U. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 64, 1986.von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 288 and 302, 1993.

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Plücker's Conoid

Cite this as:

Weisstein, Eric W. "Plücker's Conoid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PlueckersConoid.html

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