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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 interpretation. Imagine that an observer moves along the ...
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 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 ...
A plane curve with non-vanishing curvature has zero torsion at all points. Conversely, if the torsion of a regular curve with non-vanishing curvature is identically zero, then this curve belongs to a fixed plane. The curvature and the torsion of a helix are constant. Conversely, any space curve whose curvature and torsion are both constant and ...
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 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. If the surface is bow or cylinder shaped, the maximum and the minimum of these curvatures are the principal curvatures.
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.
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 .