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Arc length is the distance between two points along a section ... The advent of infinitesimal calculus led to a general formula that provides closed-form solutions in ...
Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. The first variation [ l ] is defined as the linear part of the change in the functional, and the second variation [ m ] is defined as the quadratic part.
The arc length, from the familiar geometry of a circle, is = The area a of the circular segment is equal to the area of the circular sector minus the area of the triangular portion (using the double angle formula to get an equation in terms of ):
Differential geometry takes another path: curves are represented in a parametrized form, and their geometric properties and various quantities associated with them, such as the curvature and the arc length, are expressed via derivatives and integrals using vector calculus.
A circular sector is shaded in green. Its curved boundary of length L is a circular arc. A circular arc is the arc of a circle between a pair of distinct points.If the two points are not directly opposite each other, one of these arcs, the minor arc, subtends an angle at the center of the circle that is less than π radians (180 degrees); and the other arc, the major arc, subtends an angle ...
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 ...
When the sagitta is small in comparison to the radius, it may be approximated by the formula [2] s ≈ l 2 8 r . {\displaystyle s\approx {\frac {l^{2}}{8r}}.} Alternatively, if the sagitta is small and the sagitta, radius, and chord length are known, they may be used to estimate the arc length by the formula
These formulas can be derived from the special case of arc-length parametrization in the following way. The above condition on the parametrisation imply that the arc length s is a differentiable monotonic function of the parameter t , and conversely that t is a monotonic function of s .
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