Estimating stretching energy from gaussian curvature
gaussian curvature cg ¼ c1c2
1/c2
1/c1
length contracts
by factor g È c1c2 x2
x
in general
ÒchargeÓ
ÒpotentialÓ
Example: stretching energy of a
ridge
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-
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-
-
-
+
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˜ c1c2 =
0
surface
g È c1c2 w2
stretching
energy
~ ˜ g2 ~ (wX) (c1c2)2 w4
X
w
typical gaussian curvature: c2 È 1/ w ; c1 È -w / X2 ; c1c2 È -1/X2
gives stretching energy ~ w5 /X,
as derived previously
(Gauss-Bonnet theorem)