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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 ...
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 ...
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.
Jean Frédéric Frenet (French:; 7 February 1816 – 12 June 1900) was a French mathematician, astronomer, and meteorologist. He was born and died in Périgueux , France. He is best known for being an independent co-discoverer of the Frenet–Serret formulas .
The Darboux vector provides a concise way of interpreting curvature κ and torsion τ geometrically: curvature is the measure of the rotation of the Frenet frame about the binormal unit vector, whereas torsion is the measure of the rotation of the Frenet frame about the tangent unit vector. [2]
In affine geometry, the Frenet–Serret frame is no longer well-defined, but it is possible to define another canonical moving frame along a curve which plays a similar decisive role. The theory was developed in the early 20th century, largely from the efforts of Wilhelm Blaschke and Jean Favard .