“second fundamental form”的概念、定义、翻译、参考文献-科学参考

second fundamental form

Mathematics

For a parametrized surface r(u,v), smooth at a point p, so that r_u = ∂r/∂u and r_v = ∂r/∂v is a basis for the tangent space, the second fundamental form is given by

$α r_{u} + β r_{v} \mapsto L α^{2} + 2 M α β + N β^{2}$

where L = r_uu·n, M = r_uv·n, N = r_vv·n, and n denotes the unit normal. Unlike the first fundamental form, which relates to the intrinsic metric properties of the surface, the second fundamental form relates to the embedding of the surface in ℝ³. For example, the cylinder and plane are locally isometric but have different second fundamental forms. The two fundamental forms must satisfy certain compatibility equations (one of which is given by the Theorema Egregium). Two surfaces in ℝ³ with the same fundamental forms are related by an isometry of ℝ³.

单词	second fundamental form
释义	second fundamental form Mathematics For a parametrized surface r(u,v), smooth at a point p, so that r_u = ∂r/∂u and r_v = ∂r/∂v is a basis for the tangent space, the second fundamental form is given by $α r_{u} + β r_{v} \mapsto L α^{2} + 2 M α β + N β^{2}$ where L = r_uu·n, M = r_uv·n, N = r_vv·n, and n denotes the unit normal. Unlike the first fundamental form, which relates to the intrinsic metric properties of the surface, the second fundamental form relates to the embedding of the surface in ℝ³. For example, the cylinder and plane are locally isometric but have different second fundamental forms. The two fundamental forms must satisfy certain compatibility equations (one of which is given by the Theorema Egregium). Two surfaces in ℝ³ with the same fundamental forms are related by an isometry of ℝ³.
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