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Animation of the curvature and the acceleration vector T′(s) As in the case of curves in two dimensions, the curvature of a regular space curve C in three dimensions (and higher) is the magnitude of the acceleration of a particle moving with unit speed along a curve. Thus if γ(s) is the arc-length parametrization of C then the unit tangent ...
Loosely speaking, the vector functions representing C and S agree together with their first and second derivatives at P. If the derivative of the curvature with respect to s is nonzero at P then the osculating circle crosses the curve C at P. Points P at which the derivative of the curvature is zero are called vertices.
A normal vector of length one is called a unit normal vector. A curvature vector is a normal vector whose length is the curvature of the object. Multiplying a normal vector by −1 results in the opposite vector, which may be used for indicating sides (e.g., interior or exterior).
Radius of curvature and center of curvature. In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or ...
The product k 1 k 2 of the two principal curvatures is the Gaussian curvature, K, and the average (k 1 + k 2)/2 is the mean curvature, H. If at least one of the principal curvatures is zero at every point, then the Gaussian curvature will be 0 and the surface is a developable surface. For a minimal surface, the mean curvature is zero at every ...
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
A space curve; the vectors T, N, B; and the osculating plane spanned by T and N. In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space, or the geometric properties of the curve itself irrespective of any motion.
where κ n−1 is last Frenet curvature (the torsion of the curve) and sgn is the signum function. The minimum total absolute curvature of any three-dimensional curve representing a given knot is an invariant of the knot. This invariant has the value 2 π for the unknot, but by the Fáry–Milnor theorem it is at least 4 π for any other knot. [2]