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Ding-Dong Surface


Ding-DongSurface

The ding-dong surface is the cubic surface of revolution given by the equation

 x^2+y^2=(1-z)z^2
(1)

(Hauser 2003) that is closely related to the kiss surface.

The surface can be represented in parametric form as

x(u,v)=avsqrt(1-v)cosu
(2)
y(u,v)=avsqrt(1-v)sinu
(3)
z(u,v)=av
(4)

for u in [0,2pi) and v in (-infty,1). In this parametrization, the coefficients of the first fundamental form are

E=a^2v^2(1-v)
(5)
F=0
(6)
G=a^2(8+(9v-16)v)/(4(1-v))
(7)

and of the second fundamental form are

e=(2a(v-1)|v|)/(sqrt((9v-16)v+8))
(8)
f=0
(9)
g=((4-3v)asgn(v))/(2(v-1)sqrt((9v-16)v+8)).
(10)

The Gaussian and mean curvatures are given by

K=(4(4-3v))/(a^2v[8+v(9v-16)]^2)
(11)
H=-(2[4+3(v-2)v]|v|)/(av^2[8+v(9v-16)]^(3/2)).
(12)

The Gaussian curvature can be given implicitly by

 K(x,y,z)=-(4(-4+3z))/(z(8-16z+9z^2)^2).
(13)

The surface area and volume enclosed by the upper teardrop are

S=1/(243)pia^2[21+48sqrt(2)+16ln2+64ln(1+sqrt(2))]
(14)
V=1/(12)pia^3.
(15)

It has centroid at (0,0,3/5a), and moment of inertia tensor

 I=[3/7Ma^2 0 0; 0 3/7Ma^2 0; 0 0 2/(35)Ma^2]
(16)

for a solid teardrop with uniform density and mass M.


See also

Cubic Surface, Kiss Surface, Pear-Shaped Curve, Teardrop Curve

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References

Hauser, H. "Gallery of Singular Algebraic Surfaces: Dingdong." https://homepage.univie.ac.at/herwig.hauser/gallery.html.Hauser, H. "The Hironaka Theorem on Resolution of Singularities (Or: A Proof We Always Wanted to Understand)." Bull. Amer. Math. Soc. 40, 323-403, 2003.

Cite this as:

Weisstein, Eric W. "Ding-Dong Surface." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Ding-DongSurface.html

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