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In most cases, including even simple curves, there are no closed-form solutions for arc length and numerical integration is necessary. Numerical integration of the arc length integral is usually very efficient. For example, consider the problem of finding the length of a quarter of the unit circle by numerically integrating the arc length integral.
Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures ...
A parametric C r-curve or a C r-parametrization is a vector-valued function: that is r-times continuously differentiable (that is, the component functions of γ are continuously differentiable), where , {}, and I is a non-empty interval of real numbers.
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 ...
Important quantities in the Whewell equation. The Whewell equation of a plane curve is an equation that relates the tangential angle (φ) with arc length (s), where the tangential angle is the angle between the tangent to the curve at some point and the x-axis, and the arc length is the distance along the curve from a fixed point.
The semicubical parabola was discovered in 1657 by William Neile who computed its arc length. Although the lengths of some other non-algebraic curves including the logarithmic spiral and cycloid had already been computed (that is, those curves had been rectified ), the semicubical parabola was the first algebraic curve (excluding the line and ...
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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