Kiss Surface

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KissSurface

The kiss surface is the quintic surface of revolution given by the equation

x^2+y^2=(1-z)z^4

(1)

that is closely related to the ding-dong surface. It is so named because the shape of the lower portion resembles that of a Hershey's Chocolate Kiss.

It can be represented parametrically as

x(u,v) = [画像:av^2sqrt((1-v)/2)cosu]

(2)

y(u,v) = [画像:av^2sqrt((1-v)/2)sinu]

(3)

z(u,v) = av.

(4)

The coefficients of the first fundamental form are

E = 1/2a^2(1-v)v^4

(5)

F =

(6)

G = [画像:(a^2(8-8v+16v^2-40v^3+25v^4))/(8(v-1))]

(7)

and of the second fundamental form are

e = [画像:(2a(v-1)v^2)/(sqrt(8-8v+16v^2-40v^3+25v^4))]

(8)

f =

(9)

g = [画像:(a(8-24v+15v^2))/(2(1-v)sqrt(8-8v+16v^2-40v^3+25v^4)).]

(10)

The Gaussian and mean curvatures are given by

K = [画像:-(16(8-24v+15v^2))/(a^2v^2(8-8v+16v^2-40v^3+25v^4)^2)]

(11)

H = [画像:-(4(4-4v+4v^2-8v^3+5v^4))/(av^2(8-8v+16v^2-40v^3+25v^4)^(3/2)).]

(12)

The Gaussian curvature can be given implicitly by

[画像: K(x,y,z)=(16(8-24z+15z^2))/(z^2(8-8z+16z^2-40z^3+25z^4)^2). ]

(13)

The surface area and volume enclosed of the top teardrop are given by

S = 0.755023...a^2

(14)

V = 1/(60)pia^3.

(15)

Its centroid is at (0,0,5/7a) and the moment of inertia tensor is

[画像: I=[(251)/(462)Ma^2 0 0; 0 (251)/(462)M 0; 0 0 1/(66)Ma^2] ]

(16)

for a solid kiss with uniform density and mass M.

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References

Nordstrand, T. "Surfaces." http://jalape.no/math/surfaces.htm.

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Kiss Surface

Cite this as:

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

Kiss Surface

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