“Serret-Frenet formulae”的概念、定义、习题解题方法及翻译-数学辞典

单词

Serret-Frenet formulae

释义

Serret-Frenet formulae

Three differential equations describing and governing how a curve in three dimensions evolves. For a curve r(s), parametrized by arc length s, the tangent vector t = dr/ds is a unit vector. Necessarily, dt/ds is perpendicular to t, so dt/ds = κn, where n is a unit vector called the normal vector, and κ>0 is the curvature. The unit vector b = t×n is the binormal vector. Then {t,n,b} is an orthonormal basis for each s. The Serret-Frenet formulae state:

where τ denotes torsion. If the torsion of a curve is 0, then the curve is planar. κ(s) and τ(s) determine a curve up to isometry.

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