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For a rectifiable curve these approximations don't get arbitrarily large (so the curve has a finite length). If a curve can be parameterized as an injective and continuously differentiable function (i.e., the derivative is a continuous function) f : [ a , b ] → R n {\displaystyle f\colon [a,b]\to \mathbb {R} ^{n}} , then the curve is ...
That is, when a family of minimizing curves is constructed, the values of the optical length satisfy the characteristic equation corresponding the wave equation. Hence, solving the associated partial differential equation of first order is equivalent to finding families of solutions of the variational problem.
has a length equal to one and is thus a unit tangent vector. If the curve is twice differentiable, that is, if the second derivatives of x and y exist, then the derivative of T(s) exists. This vector is normal to the curve, its length is the curvature κ(s), and it is oriented toward the center of curvature. That is,
Therefore an intrinsic equation defines the shape of the curve without specifying its position relative to an arbitrarily defined coordinate system. The intrinsic quantities used most often are arc length s {\displaystyle s} , tangential angle θ {\displaystyle \theta } , curvature κ {\displaystyle \kappa } or radius of curvature , and, for 3 ...
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 ...
Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the synthetic approach .
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.
A space curve is a curve for which is at least three-dimensional; a skew curve is a space curve which lies in no plane. These definitions of plane, space and skew curves apply also to real algebraic curves , although the above definition of a curve does not apply (a real algebraic curve may be disconnected ).
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