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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 ...
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 other words, if γ 1 (t) and γ 2 (t) are two curves in such that for any t, the two principal normals N 1 (t), N 2 (t) are equal, then γ 1 and γ 2 are Bertrand curves, and γ 2 is called the Bertrand mate of γ 1. We can write γ 2 (t) = γ 1 (t) + r N 1 (t) for some constant r. [1]
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.
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.
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.
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 .