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In differential geometry, the Frenet–Serret 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 Frédéric 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–Serret frame or TNB frame, together form an orthonormal basis spanning ℝ3 and are defined as follows: * T is the unit vector tangent to the curve, pointing in the direction of motion. * N is the normal unit vector, the derivative of T with respect to the arclength parameter of the curve, divided by its length. * B is the binormal unit vector, the cross product of T and N. The Frenet–Serret formulas are: : where ''d''/''ds'' is the derivative with respect to arclength, ''κ'' is the curvature, and ''τ'' is the torsion of the curve. The two scalars ''κ'' and ''τ'' effectively define the curvature and torsion of a space curve. The associated collection, T, N, B, ''κ'', and ''τ'', is called the Frenet–Serret 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. ==Definitions== Let r(t) be a curve in Euclidean space, representing the position vector of the particle as a function of time. The Frenet–Serret formulas apply to curves which are ''non-degenerate'', which roughly means that they have nonzero curvature. More formally, in this situation the velocity vector r′(t) and the acceleration vector r′′(t) are required not to be proportional. Let ''s(t)'' represent the arc length which the particle has moved along the curve. The quantity ''s'' is used to give the curve traced out by the trajectory of the particle a natural parametrization by arc length, since many different particle paths may trace out the same geometrical curve by traversing it at different rates. In detail, ''s'' is given by : Moreover, since we have assumed that r′ ≠ 0, it follows that ''s''(''t'') is a strictly monotonically increasing function. Therefore, it is possible to solve for ''t'' as a function of ''s'', and thus to write r(''s'') = r(''t''(''s'')). The curve is thus parametrized in a preferred manner by its arc length. With a non-degenerate curve r(''s''), parameterized by its arc length, it is now possible to define the Frenet–Serret frame (or TNB frame): * The tangent unit vector T is defined as :: * The normal unit vector N is defined as :: * The binormal unit vector B is defined as the cross product of T and N: :: From equation (2) it follows, since T always has unit magnitude, that N is always perpendicular to T. From equation (3) it follows that B is always perpendicular to both T and N. Thus, the three unit vectors T, N, and B are all perpendicular to each other. The Frenet–Serret formulas are: : where is the curvature and is the torsion. The Frenet–Serret formulas are also known as ''Frenet–Serret theorem'', and can be stated more concisely using matrix notation: : This matrix is skew-symmetric. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Frenet–Serret formulas」の詳細全文を読む スポンサード リンク
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