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The sector contour used to calculate the limits of the Fresnel integrals. This can be derived with any one of several methods. One of them [5] uses a contour integral of the function around the boundary of the sector-shaped region in the complex plane formed by the positive x-axis, the bisector of the first quadrant y = x with x ≥ 0, and a circular arc of radius R centered at the origin.
An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from or wrapped around the curve. [1] The evolute of an involute is the original curve. It is generalized by the roulette family of curves. That is, the involutes of a curve are the roulettes of the curve generated by a straight line.
Thus the length of a curve is a non-negative real number. Usually no curves are considered which are partly spacelike and partly timelike. In theory of relativity, arc length of timelike curves (world lines) is the proper time elapsed along the world line, and arc length of a spacelike curve the proper distance along the curve.
Curve fitting [1] [2] is the process ... it is effective to fit each of its coordinates as a separate function of arc length; ... Igor Pro, MLAB, Maple, MATLAB, TK ...
A double-end Euler spiral. The curve continues to converge to the points marked, as t tends to positive or negative infinity. An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). This curve is also referred to as a clothoid or Cornu spiral.
However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends upon the material of the medium.
Dubins proved his result in 1957. In 1974 Harold H. Johnson proved Dubins' result by applying Pontryagin's maximum principle. [4] In particular, Harold H. Johnson presented necessary and sufficient conditions for a plane curve, which has bounded piecewise continuous curvature and prescribed initial and terminal points and directions, to have minimal length.
A space curve; the vectors T, N, B; and the osculating plane spanned by T and N. In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space, or the geometric properties of the curve itself irrespective of any motion.