Arc length

The definition of arc length for curve is analogous to the definition of the entire variation of a real-valued function.

The lengths of the successive approximations do not decrease and will gradually increase-possibly indefinitely, but for the smooth curves this tends to a limit as the lengths of segments becomes arbitrarily small. A curve in the plane and can be approximated by connecting a finite number of points on the curve using line segments to create a polygonal path.

If the curve is not a polygonal path already, better approximations to the curve can be obtained by following the shape of the curve increasingly more closely. For some curves there is a smallest number L that is the upper bound on length of any polygonal approximation. If this type of a number exists, then curve is said to be rectifiable and the curve is defined to have arc length L.

The surface of revolution is the surface in Euclidean space created by rotating a curve around a straight line in its plane (the axis). A curve can be parameterized in a number of ways. So the arc length is an intrinsic property of the curve, meaning that it does not depend on the choice of parameterization.

A surface of the revolution is a surface created by rotating a curve lying on some plane around the straight line which lies on the same plane. A circle which is rotated about its diameter generates a sphere and if the circle is rotated about a coplanar axis other than the diameter it generates a torus. The examples of surfaces generated by the straight line are cylindrical and conical surfaces when the line is coplanar with axis, along with the hyperboloids of one sheet when the line is skew to the axis. The total length of the entire arc can then be obtained by adding up all the lengths of small arc pieces in the Riemann sum sense under a limiting process.

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