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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.
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]
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
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 ...
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 ...
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).
A space curve, Frenet–Serret frame, and the osculating plane (spanned by T and N). In mathematics, particularly in differential geometry, an osculating plane is a plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second order of contact at the point.
The normal section of a surface at a particular point is the curve produced by the intersection of that surface with a normal plane. [1] [2] [3]The curvature of the normal section is called the normal curvature.