“Serret-Frenet formulae”的概念、定义、翻译、参考文献-科学参考

Serret-Frenet formulae

Mathematics

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:

$\frac{d t}{d s} = κ n, \frac{d n}{d s} = - κ t + τ b, \frac{d b}{d s} = - τ n,$

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.

单词	Serret-Frenet formulae
释义	Serret-Frenet formulae Mathematics 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: $\frac{d t}{d s} = κ n, \frac{d n}{d s} = - κ t + τ b, \frac{d b}{d s} = - τ n,$ 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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