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The first Frenet-Serret formula holds by the definition of the normal N and the curvature κ, and the third Frenet-Serret formula holds by the definition of the torsion τ. Thus what is needed is to show the second Frenet-Serret formula. Since T, N, B are orthogonal unit vectors with B = T × N, one also has T = N × B and N = B × T.
An illustration of the Frenet frame for a point on a space curve. T is the unit tangent, P the unit normal, and B the unit binormal. A Frenet frame is a moving reference frame of n orthonormal vectors e i (t) which are used to describe a curve locally at each point γ(t). It is the main tool in the differential geometric treatment of curves ...
The Frenet–Serret frame is a moving frame defined on a curve which can be constructed purely from the velocity and acceleration of the curve. [2] The Frenet–Serret frame plays a key role in the differential geometry of curves, ultimately leading to a more or less complete classification of smooth curves in Euclidean space up to congruence. [3]
Frenet–Serret formulas; Curves in differential geometry; Line element; Curvature; Radius of curvature; Osculating circle; Curve; Fenchel's theorem; Differential ...
The expression of the curvature In terms of arc-length parametrization is essentially the first Frenet–Serret formula ′ = (), where the primes refer to the derivatives with respect to the arc length s, and N(s) is the normal unit vector in the direction of T′(s).
which can be derived from Equation (1) by means of the Frenet-Serret theorem (or vice versa). Let a rigid object move along a regular curve described parametrically by β(t). This object has its own intrinsic coordinate system. As the object moves along the curve, let its intrinsic coordinate system keep itself aligned with the curve's Frenet ...
Animation of the torsion and the corresponding rotation of the binormal vector. Let r be a space curve parametrized by arc length s and with the unit tangent vector T.If the curvature κ of r at a certain point is not zero then the principal normal vector and the binormal vector at that point are the unit vectors
In the case of the Frenet–Serret frame, the structural equations are precisely the Frenet–Serret formulas, and these serve to classify curves completely up to Euclidean motions. The general case is analogous: the structural equations for an adapted system of frames classifies arbitrary embedded submanifolds up to a Euclidean motion.