. . . . . . . "1066906286"^^ . . . . . . "In differential geometry, the Frenet\u2013Serret formulas describe the kinematic properties of a particle moving along a continuous, differentiable curve in three-dimensional Euclidean space R3, or the geometric properties of the curve itself irrespective of any motion. More specifically, the formulas describe the derivatives of the so-called tangent, normal, and binormal unit vectors in terms of each other. The formulas are named after the two French mathematicians who independently discovered them: Jean Fr\u00E9d\u00E9ric Frenet, in his thesis of 1847, and Joseph Alfred Serret, in 1851. Vector notation and linear algebra currently used to write these formulas were not yet in use at the time of their discovery."@en . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "666987"^^ . "Frenet\u2013Serret formulas"@en . . . . . . . . . . . . . . . . . "32934"^^ . . . . . . . . . "In differential geometry, the Frenet\u2013Serret formulas describe the kinematic properties of a particle moving along a continuous, differentiable curve in three-dimensional Euclidean space R3, or the geometric properties of the curve itself irrespective of any motion. More specifically, the formulas describe the derivatives of the so-called tangent, normal, and binormal unit vectors in terms of each other. The formulas are named after the two French mathematicians who independently discovered them: Jean Fr\u00E9d\u00E9ric Frenet, in his thesis of 1847, and Joseph Alfred Serret, in 1851. Vector notation and linear algebra currently used to write these formulas were not yet in use at the time of their discovery. The tangent, normal, and binormal unit vectors, often called T, N, and B, or collectively the Frenet\u2013Serret frame or TNB frame, together form an orthonormal basis spanning R3 and are defined as follows: \n* T is the unit vector tangent to the curve, pointing in the direction of motion. \n* N is the normal unit vector, the derivative of T with respect to the arclength parameter of the curve, divided by its length. \n* B is the binormal unit vector, the cross product of T and N. The Frenet\u2013Serret formulas are: where d/ds is the derivative with respect to arclength, \u03BA is the curvature, and \u03C4 is the torsion of the curve. The two scalars \u03BA and \u03C4 effectively define the curvature and torsion of a space curve. The associated collection, T, N, B, \u03BA, and \u03C4, is called the Frenet\u2013Serret apparatus. Intuitively, curvature measures the failure of a curve to be a straight line, while torsion measures the failure of a curve to be planar."@en .