Arc length

Arc length is the distance between two points along a section of a curve. Arc length is the distance between two points along a section of a curve. Determining the length of an irregular arc segment is also called rectification of a curve. The advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases. A curve in the plane can be approximated by connecting a finite number of points on the curve using line segments to create a polygonal path. Since it is straightforward to calculate the length of each linear segment (using the Pythagorean theorem in Euclidean space, for example), the total length of the approximation can be found by summing the lengths of each linear segment; that approximation is known as the (cumulative) chordal distance. If the curve is not already a polygonal path, using a progressively larger number of segments of smaller lengths will result in better approximations. The lengths of the successive approximations will not decrease and may keep increasing indefinitely, but for smooth curves they will tend to a finite limit as the lengths of the segments get arbitrarily small. For some curves there is a smallest number L {displaystyle L} that is an upper bound on the length of any polygonal approximation. These curves are called rectifiable and the number L {displaystyle L} is defined as the arc length. Let f : [ a , b ] → R n {displaystyle fcolon o mathbb {R} ^{n}} be a continuously differentiable function. The length of the curve defined by f {displaystyle f} can be defined as the limit of the sum of line segment lengths for a regular partition of [ a , b ] {displaystyle } as the number of segments approaches infinity. This means where t i = a + i ( b − a ) / N = a + i Δ t {displaystyle t_{i}=a+i(b-a)/N=a+iDelta t} for i = 0 , 1 , … , N . {displaystyle i=0,1,dotsc ,N.} This definition is equivalent to the standard definition of arc length as an integral: The last equality above is true because the definition of the derivative as a limit implies that there is a positive real function δ ( ϵ ) {displaystyle delta (epsilon )} of positive real ϵ {displaystyle epsilon } such that Δ t < δ ( ϵ ) {displaystyle Delta t<delta (epsilon )} implies | | f ( t i ) − f ( t i − 1 ) Δ t | − | f ′ ( t i ) | | < ϵ . {displaystyle left|{igg |}{frac {f(t_{i})-f(t_{i-1})}{Delta t}}{igg |}-{Big |}f'(t_{i}){Big |} ight|<epsilon .} This means has absolute value less than ϵ ( b − a ) {displaystyle epsilon (b-a)} for N > ( b − a ) / δ ( ϵ ) . {displaystyle N>(b-a)/delta (epsilon ).} This means that in the limit N → ∞ , {displaystyle N ightarrow infty ,} the left term above equals the right term, which is just the Riemann integral of | f ′ ( t ) | {displaystyle |f'(t)|} on [ a , b ] . {displaystyle .} This definition of arc length shows that the length of a curve f : [ a , b ] → R n {displaystyle f: ightarrow mathbb {R} ^{n}} continuously differentiable on [ a , b ] {displaystyle } is always finite. In other words, the curve is always rectifiable.