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A curve may have equivalent parametrizations when there is a continuous increasing monotonic function relating the parameter of one curve to the parameter of the other. When there is a decreasing continuous function relating the parameters, then the parametric representations are opposite and the orientation of the curve is reversed. [1] [2]
If t = s is the natural parameter, then the tangent vector has unit length. The formula simplifies: = ′ (). The unit tangent vector determines the orientation of the curve, or the forward direction, corresponding to the increasing values of the parameter.
In quaternionic formalism the choice of an orientation of the space corresponds to order of multiplication: ij = k but ji = −k. If one reverses the orientation, then the formula above becomes p ↦ q −1 p q, i.e., a unit q is replaced with the conjugate quaternion – the same behaviour as of axial 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.
This formula shows how to calculate the curl of F in any coordinate system, and how to extend the curl to any oriented three-dimensional Riemannian manifold. Since this depends on a choice of orientation, curl is a chiral operation. In other words, if the orientation is reversed, then the direction of the curl is also reversed.
This curve will in general have different curvatures for different normal planes at p. The principal curvatures at p, denoted k 1 and k 2, are the maximum and minimum values of this curvature. Here the curvature of a curve is by definition the reciprocal of the radius of the osculating circle. The curvature is taken to be positive if the curve ...
Because this equation holds for all vectors, p, one concludes that every rotation matrix, Q, satisfies the orthogonality condition, Q T Q = I . {\displaystyle Q^{\mathsf {T}}Q=I.} Rotations preserve handedness because they cannot change the ordering of the axes, which implies the special matrix condition,
The signed curvature is not defined, as it depends on an orientation of the curve that is not provided by the implicit equation. Note that changing F into – F would not change the curve defined by F ( x , y ) = 0 , but it would change the sign of the numerator if the absolute value were omitted in the preceding formula.