Ding-Dong Surface

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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 =

(6)

G = [画像:a^2(8+(9v-16)v)/(4(1-v))]

(7)

and of the second fundamental form are

e = [画像:(2a(v-1)|v|)/(sqrt((9v-16)v+8))]

(8)

f =

(9)

g = [画像:((4-3v)asgn(v))/(2(v-1)sqrt((9v-16)v+8)).]

(10)

The Gaussian and mean curvatures are given by

K = [画像:(4(4-3v))/(a^2v[8+v(9v-16)]^2)]

(11)

H = [画像:-(2[4+3(v-2)v]|v|)/(av^2[8+v(9v-16)]^(3/2)).]

(12)

The Gaussian curvature can be given implicitly by

[画像: 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

[画像: 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.

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References

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.Hauser, H. "Gallery of Singular Algebraic Surfaces: Dingdong." https://homepage.univie.ac.at/herwig.hauser/gallery.html.

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Ding-Dong Surface

Cite this as:

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

Ding-Dong Surface

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