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Arc length s of a logarithmic spiral as a function of its parameter θ. Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a focus of calculus.
Those with k > 0 are ellipses, those with k = 0 are parabolae, and those with k < 0 are hyperbolae. The usual Euclidean curvature of a curve at a point is the curvature of its osculating circle , the unique circle making second order contact (having three point contact) with the curve at the point.
The coordinate-independent definition of the square of the line element ds in an n-dimensional Riemannian or Pseudo Riemannian manifold (in physics usually a Lorentzian manifold) is the "square of the length" of an infinitesimal displacement [2] (in pseudo Riemannian manifolds possibly negative) whose square root should be used for computing curve length: = = (,) where g is the metric tensor ...
The Cesàro equation is obtained as a relation between arc length and curvature. The equation of a circle (including a line) for example is given by the equation κ ( s ) = 1 r {\displaystyle \kappa (s)={\tfrac {1}{r}}} where s {\displaystyle s} is the arc length, κ {\displaystyle \kappa } the curvature and r {\displaystyle r} the radius of ...
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 degree of curvature is based on 100 units of arc length, the conversion between degree of curvature and radius is Dr = 18000/π ≈ 5729.57795, where D is degree and r is radius. Since rail routes have very large radii, they are laid out in chords, as the difference to the arc is inconsequential; this made work easier before electronic ...
When the relation is a function, so that tangential angle is given as a function of arc length, certain properties become easy to manipulate. In particular, the derivative of the tangential angle with respect to arc length is equal to the curvature. Thus, taking the derivative of the Whewell equation yields a Cesàro equation for the same curve.
The limaçon trisectrix specified as the polar equation = (+ ), where a > 0. When a < 0, the resulting curve is the reflection of this curve with respect to the line = / As a function, r has a period of 2π. The inner and outer loops of the curve intersect at the pole.