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Arc length is the distance between two points along a section of a curve. Determining the length of an irregular arc segment by approximating the arc segment as connected (straight) line segments is also called curve rectification .
A diagram illustrating great-circle distance (drawn in red) between two points on a sphere, P and Q. Two antipodal points, u and v are also shown. The great-circle distance, orthodromic distance, or spherical distance is the distance between two points on a sphere, measured along the great-circle arc between them. This arc is the shortest path ...
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 ...
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 ...
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 ...
In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points F 1 and F 2, known as foci, at distance 2c from each other as the locus of points P so that PF 1 ·PF 2 = c 2. The curve has a shape similar to the numeral 8 and to the ∞ symbol.
The essential property of the tractrix is constancy of the distance between a point P on the curve and the intersection of the tangent line at P with the asymptote of the curve. The tractrix might be regarded in a multitude of ways: It is the locus of the center of a hyperbolic spiral rolling (without skidding) on a straight line.