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The length of the curve is given by the formula = | ′ | where | ′ | is the Euclidean norm of the tangent vector ′ to the curve. To justify this formula, define the arc length as limit of the sum of linear segment lengths for a regular partition of [ a , b ] {\displaystyle [a,b]} as the number of segments approaches infinity.
Also, let Q = (x 1, y 1) be any point on this line and n the vector (a, b) starting at point Q. The vector n is perpendicular to the line, and the distance d from point P to the line is equal to the length of the orthogonal projection of on n. The length of this projection is given by:
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 equator of the unit sphere is a parametrized curve given by ((), ()) = (,) with t ranging from 0 to 2 π. The line element may be used to calculate the length of this curve. The line element may be used to calculate the length of this curve.
It can be extended to infinite-dimensional vector spaces as the L 2 norm or L 2 distance. [25] The Euclidean distance gives Euclidean space the structure of a topological space, the Euclidean topology, with the open balls (subsets of points at less than a given distance from a given point) as its neighborhoods. [26]
T is the unit vector tangent to the curve, pointing in the direction of motion. N is the normal unit vector, the derivative of T with respect to the arclength parameter of the curve, divided by its length. B is the binormal unit vector, the cross product of T and N.
In the case of a space curve, the radius of curvature is the length of the curvature vector. In the case of a plane curve, then R is the absolute value of [3] | | =, where s is the arc length from a fixed point on the curve, φ is the tangential angle and κ is the curvature.
The arc length (length of a line segment) defined by a polar function is found by the integration over the curve r(φ). Let L denote this length along the curve starting from points A through to point B, where these points correspond to φ = a and φ = b such that 0 < b − a < 2π.