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The Frenet–Serret frame moving along a helix in space. The Frenet–Serret frame consisting of the tangent T, normal N, and binormal B collectively forms an orthonormal basis of 3-space. At each point of the curve, this attaches a frame of reference or rectilinear coordinate system (see image). The Frenet–Serret formulas admit a kinematic ...
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 torsion tensor thus is related to, although distinct from, the torsion of a curve, as it appears in the Frenet–Serret formulas: the torsion of a connection measures a dislocation of a developed curve out of its plane, while the torsion of a curve is also a dislocation out of its osculating plane.
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
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] The Frenet–Serret formulas show that there is a pair of functions defined on the curve, the torsion and curvature, which are obtained by ...
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 ...
From just the curvature and torsion, the vector fields for the tangent, normal, and binormal vectors can be derived using the Frenet–Serret formulas. Then, integration of the tangent field (done numerically, if not analytically) yields the curve.
A diagonalizing frame a 1, a 2, a 3 consists of the normal vector a 3, and two principal directions a 1 and a 2. This is called a Darboux frame on the surface. The frame is canonically defined (by an ordering on the eigenvalues, for instance) away from the umbilics of the surface.