Let be a regular surface with points in the tangent space of . For , the second fundamental form is the symmetric bilinear form on the tangent space ,
(1)
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where is the shape operator. The second fundamental form satisfies
(2)
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for any nonzero tangent vector.
The second fundamental form is given explicitly by
(3)
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where
(4)
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(5)
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(6)
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and are the direction cosines of the surface normal. The second fundamental form can also be written
(7)
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(8)
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(9)
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(10)
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(11)
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(12)
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(13)
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(14)
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where is the normal vector, is a regular patch, and and are the partial derivatives of with respect to parameters and , respectively, or
(15)
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(16)
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(17)
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