Informally, such curves are said to have infinite length. There are continuous curves on which every arc (other than a single-point arc) has infinite length. An example of such a curve is the koch curve. Another example of a curve with infinite length is the graph of the function defined by f ( x ) x sin(1/ x ) for any open set with 0 as one of its delimiters and f (0). Sometimes the hausdorff dimension and hausdorff measure are used to quantify the size of such curves. Generalization to (pseudo-)Riemannian manifolds edit let Mdisplaystyle m be a (pseudo-)Riemannian manifold, γ:0,1Mdisplaystyle gamma :0,1rightarrow m a curve in Mdisplaystyle m and gdisplaystyle g the (pseudo-) metric tensor. The length of γdisplaystyle gamma is defined to be ell (gamma )int _01sqrt pm g(gamma t gamma t dt, where γ(t)Tγ(t)Mdisplaystyle gamma t)in T_gamma (t)M is the tangent vector of γdisplaystyle gamma.
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As an example of his method, he determined the essay arc length of a semicubical parabola, which required finding the area under a parabola. 6 In 1660, fermat published a more general theory containing the same result in his de linearum curvarum cum lineis rectis comparatione dissertatio geometrica (Geometric dissertation on curved lines in comparison with straight lines). 7 Fermat's method of determining arc length building on his previous work with tangents, fermat used the curve yx3/2displaystyle yx3/2, whose tangent the at x a had a slope of 32a1/2displaystyle textstyle 3 over 2a1/2 so the tangent line would have the equation y32a1/2(xa)f(a).displaystyle ytextstyle. Next, he increased a by a small amount to a ε, making segment ac a relatively good approximation for the length of the curve from A. To find the length of the segment ac, he used the pythagorean theorem : varepsilon 29 over 4avarepsilon 2 textstyle varepsilon 2left(19 over 4aright)endaligned which, when solved, yields acε194a.displaystyle actextstyle varepsilon sqrt 19 over. In order to approximate the length, fermat would sum up a sequence of short segments. Curves with infinite length edit see also: coastline paradox The koch curve. The graph of x sin(1/ x ). As mentioned above, some curves are non-rectifiable. That is, there is no upper bound on the lengths of polygonal approximations; the length can be made arbitrarily large.
By using plan more segments, and by decreasing the length of each segment, they were able to obtain a more and more accurate approximation. In particular, by inscribing a polygon of many sides in a circle, they were able to find approximate values. 17th century edit In the 17th century, the method of exhaustion led to the rectification by geometrical methods of several transcendental curves : the logarithmic spiral by evangelista torricelli in 1645 (some sources say john Wallis in the 1650s the cycloid by Christopher Wren. In 1659, wallis credited William neile 's discovery of the first rectification of a nontrivial algebraic curve, the semicubical parabola. 5 The accompanying figures appear on page 145. On page 91, william neile is mentioned as Gulielmus Nelius. Integral form edit before the full formal development of calculus, the basis for the modern integral form for arc length was independently discovered by hendrik van heuraet and pierre de fermat. In 1659 van heuraet published a construction showing that the problem of determining arc length could be transformed into the problem of determining the area under a curve (i.e., an integral).
Those definitions of the metre and the nautical mile have been superseded by more precise ones, but the original definitions are still accurate enough for conceptual purposes and some calculations. For example, they imply that supermarket one kilometre is exactly.54 nautical miles. Using official modern definitions, one nautical mile is exactly.852 kilometres, 3 which implies that 1 kilometre is about.53995680 nautical miles. 4 This modern ratio differs from the one calculated from the original definitions by less than one part in 10,000. Length of an arc of a parabola edit for a calculation of the length of a parabolic arc, see parabola Length of an arc of a parabola. Historical methods edit Antiquity edit for much of the history of mathematics, even the greatest thinkers considered it impossible to compute the length of an irregular arc. Although Archimedes had pioneered a way of finding the area beneath a curve with his " method of exhaustion few believed it was even possible for curves to have definite lengths, as do straight lines. The first ground was broken in this field, as it often has been in calculus, by approximation. People began to inscribe polygons within the curves and compute the length of the sides for a somewhat accurate measurement of the length.
For an arbitrary circular arc: If θdisplaystyle theta is in radians then srθ. This is a definition of the radian. If θdisplaystyle theta is in degrees, then sπrθ180,displaystyle sfrac pi rtheta 180, which is the same as sCθ360.displaystyle sfrac Ctheta 360. If θdisplaystyle theta is in grads (100 grads, or grades, or gradians are one right-angle then sπrθ200,displaystyle sfrac pi rtheta 200, which is the same as sCθ400.displaystyle sfrac Ctheta 400. If θdisplaystyle theta is in turns (one turn is a complete rotation, or 360, or 400 grads, or 2πdisplaystyle 2pi radians then sCθ. Arcs of great circles on the earth edit main article: Great-circle distance further information: geodesics on an ellipsoid Two units of length, the nautical mile and the metre (or kilometre were originally defined so the lengths of arcs of great circles on the earth's surface. The simple equation sθdisplaystyle stheta applies in the following circumstances: if sdisplaystyle s is in nautical miles, and θdisplaystyle theta is in arcminutes (160 degree or if sdisplaystyle s is in kilometres, and θdisplaystyle theta is in centigrades (1100 grad). The lengths of the distance units were chosen to make the circumference of the earth equal kilometres, or nautical miles. Those are the numbers of the corresponding angle units in one complete turn.
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Now let mathbf C (t r(t theta (t phi (t) be a curve expressed in spherical coordinates where θdisplaystyle theta is the polar angle measured from the positive zdisplaystyle z -axis and ϕdisplaystyle phi is the azimuthal angle. The mapping that transforms from spherical coordinates to rectangular coordinates is mathbf x (r,theta,phi rsin theta cos phi,rsin theta sin phi, rcos theta ). Using the chain rule again shows that D(xC)xrrxθxϕ. Displaystyle D(mathbf x circ mathbf C )mathbf x _rr'mathbf x _theta theta 'mathbf x _phi phi '. All dot products xixjdisplaystyle mathbf x _icdot mathbf x _j where idisplaystyle i and jdisplaystyle j differ are zero, so the squared the norm of this vector is (mathbf x _rcdot mathbf x _r r'2 mathbf x _theta cdot mathbf x _theta theta 2(mathbf x _phi. So for a curve expressed in spherical coordinates, the arc length is int _t_1t_2sqrt left(frac drdtright)2r2left(frac dtheta dtright)2r2sin 2theta left(frac dphi dtright)2dt. A very similar calculation shows that the arc length of a curve expressed in cylindrical coordinates is int _t_1t_2sqrt left(frac drdtright)2r2left(frac dtheta dtright)2left(frac dzdtright)2dt.
Simple cases edit Arcs of circles edit Arc lengths are denoted by s, since the latin word for length (or size) is spatium. In the following lines, rdisplaystyle r represents the radius of a circle, ddisplaystyle d is its diameter, cdisplaystyle c is its circumference, sdisplaystyle s is the length of an arc of the circle, and θdisplaystyle theta is the angle which the arc subtends at the. The distances r,d,C,displaystyle r,d, c, and sdisplaystyle s are expressed in the same units. C2πr, displaystyle C2pi r, which is the same as Cπd. This equation is a definition. If the arc is a semicircle, then sπr.
Differs from the true length of π/2displaystyle pi /2.31011 and the 16-point gaussian quadrature rule estimate. Differs from the true length by only.71013. This means it is possible to evaluate this integral to almost machine precision with only 16 integrand evaluations. Curve on a surface edit let x(u,v)displaystyle mathbf x (u,v) be a surface mapping and let C(t u(t v(t)displaystyle mathbf C (t u(t v(t) be a curve on this surface. The integrand of the arc length integral is (xC t).displaystyle (mathbf x circ mathbf c t).
Evaluating the derivative requires the chain rule for vector fields: D(xC xu xv uv)xuuxvv. Displaystyle D(mathbf x circ mathbf c mathbf x _u mathbf x _v)binom u'v'mathbf x _uu'mathbf x _vv'. The squared norm of this vector is (mathbf x _uu'mathbf x _vv cdot (mathbf x _uu'mathbf x (where gijdisplaystyle g_ij is the first fundamental form coefficient so the integrand of the arc length integral can be written as gab(ua ub)displaystyle sqrt g_ab(ua ub (where u1udisplaystyle. Other coordinate systems edit let C(t r(t θ(t)displaystyle mathbf C (t r(t theta (t) be a curve expressed in polar coordinates. The mapping that transforms from polar coordinates to rectangular coordinates is mathbf x (r,theta rcos theta, rsin theta ). The chain rule for vector fields shows that D(xC)xrrxθ. Displaystyle D(mathbf x circ mathbf C )mathbf x _rr'mathbf x _theta theta '. So the squared integrand of the arc length integral is (mathbf x_r cdot mathbf x _r r 22(mathbf x _rcdot mathbf x _theta )r'theta mathbf x _theta cdot mathbf x _theta theta 2(r 2r2(theta. So for a curve expressed in polar coordinates, the arc length is int _t_1t_2sqrt left(frac drdtright)2r2left(frac dtheta dtright)2dtint _theta (t_1)theta (t_2)sqrt left(frac drdtheta right)2r2dtheta.
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Numerical integration of the arc length integral is usually very efficient. For example, shredder consider the problem of finding the length of a quarter of the unit circle by numerically integrating the arc length integral. The upper half of the unit circle can be parameterized as y1x2.displaystyle ysqrt 1-x2. The interval x2/2,2/2displaystyle xin -sqrt 2/2,sqrt 2/2 corresponds to a quarter of the circle. Since dy/dxx/1x2displaystyle with dy/dx-x/sqrt 1-x2 and 1(dy/dx)21 1x2 displaystyle 1(dy/dx)21 1-x2 the length of a quarter of the unit circle is 2/22/211x2dx. Displaystyle int _-sqrt 2/2sqrt 2/2frac 1sqrt 1-x2dx. The 15-point gauss-Kronrod rule estimate for this integral.
A curve can be parameterized in infinitely many ways. Let φ:a,bc, ddisplaystyle varphi :a,bto c, d be any continuously differentiable bijection. Then gfφ1:c,dRndisplaystyle gfcirc varphi -1:c,dto mathbb R n is another continuously differentiable parameterization of the curve originally defined. The arc length of the curve is the same regardless of the parameterization used to define the curve: L(f)abf(t) dtabg(φ(t)φ(t) dtabg(φ(t)φ(t) dtsince φ must be non-decreasingcdg(u) duusing integration by substitutionL(g).displaystyle beginalignedL(f) int _abBig f t)Big dtint _abBig g varphi (t)varphi t)Big dt int _abBig g varphi (t)Big varphi t) dtquad textrm since varphi textrm must textrm. Displaystyle sint _absqrt 1left(frac dydxright)2dx. Curves with closed-form solutions for arc length include the catenary, circle, cycloid, logarithmic spiral, parabola, semicubical parabola and straight line. The lack essay of a closed form solution for the arc length of an elliptic arc led to the development of the elliptic integrals. Numerical integration edit In most cases, including even simple curves, there are no closed-form solutions for arc length and numerical integration is necessary.
the definition of the derivative as a limit implies that there is a positive real function δ(ϵ)displaystyle delta (epsilon ) of positive real ϵdisplaystyle epsilon such that Δt δ(ϵ)displaystyle delta t delta (epsilon ) implies leftbigg frac f(t_i)-f(t_i-1)Delta. This means sum _i1Nleftfrac f(t_i)-f(t_i-1)Delta trightDelta t-sum _i1NBig f t_i)Big Delta t has absolute value less than ϵ(ba)displaystyle epsilon (b-a) for N (ba δ(ϵ).displaystyle n (b-a delta (epsilon ). This means that in the limit N, displaystyle Nrightarrow infty, the left term above equals the right term, which is just the riemann integral of f(t)displaystyle f t) on a,b. This definition of arc length shows that the length of a curve f:a,bRndisplaystyle f:a,brightarrow mathbb R n continuously differentiable on a,bdisplaystyle a, b is always finite. In other words, the curve is always rectifiable. The definition of arc length of a smooth curve as the integral of the norm of the derivative is equivalent to the definition L(f)sup sum _i1Nbigg f(t_i)-f(t_i-1)bigg where the supremum is taken over all possible partitions at_0 t_1 dots t_N-1 t_Nb of a,b. 2 This definition is also valid if fdisplaystyle f is merely continuous, not differentiable.
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. 1, 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 paper 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 Ldisplaystyle l that is an upper bound on the length of any polygonal approximation. These curves are called rectifiable and the number Ldisplaystyle l is defined as the arc length. Definition for a smooth curve edit see also: Length of a curve let f:a,bRndisplaystyle fcolon a, bto mathbb R n be a continuously differentiable function. The length of the curve defined by fdisplaystyle f can be defined as the limit of the sum of line segment lengths for a regular partition of a,bdisplaystyle a, b as the number of segments approaches infinity. This means L(f)lim _Nto infty sum _i1Nbigg f(t_i)-f(t_i-1)bigg where tiai(ba naiδtdisplaystyle t_iai(b-a naidelta t for i0,1.
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When rectified, the curve presentation gives a straight line segment with the same length as the curve's arc length. Example: arc length s of a logarithmic spiral as a function of its parameter. Determining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves. The advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases. Contents, general approach edit, approximation by multiple linear segments, 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.