Curvature. (Com S 477/577 Notes) Yan-Bin Jia. Oct 8, 2015

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1 Cuvtue Com S 477/577 Notes Yn-Bin Ji Oct 8, 205 We wnt to find mesue of how cuved cuve is. Since this cuvtue should depend only on the shpe of the cuve, it should not be chnged when the cuve is epmetized. Futhe, the mesue of cuvtue should gee with ou intuition in simple specil cses. Stight lines themselves hve zeo cuvtue. Lge cicles should hve smlle cuvtue thn smll cicles which bend moe shply. The signed cuvtue of cuve pmetized by its c length is the te of chnge of diection of the tngent vecto. The bsolute vlue of the cuvtue is mesue of how shply the cuve bends. Cuves which bend slowly, which e lmost stight lines, will hve smll bsolute cuvtue. Cuves which swing to the left hve positive cuvtue nd cuves which swing to the ight hve negtive cuvtue. The cuvtue of the diection of od will ffect the mximum speed t which vehicles cn tvel without skidding, nd the cuvtue in the tjectoy of n iplne will ffect whethe the pilot will suffe blckout s esult of the g-foces involved. In this lectue we will pimily look t the cuvtue of plne cuves. The esults will be extended to spce cuves in the next lectue. Cuvtue To intoduce the definition of cuvtue, in this section we conside unit-speed cuve αs, whee s is the c length. The tngentil ngle φ is mesued counteclockwise fom the x-xis to the unit tngent T = α s, s shown below. αs φs x The cuvtue κ of α is the te of chnge in the diection of the tngent line t tht point with espect to c length, tht is, κ = dφ ds.

2 The bsolute cuvtue of the cuve t the point is the bsolute vlue κ. Since α hs unit speed, T T =. Diffeentiting this eqution yields T T = 0. The chnge of Ts is othogonl to the tngentil diection, so it must be long the noml diection. The cuvtue is lso defined to mesue the tuning of Ts long the diection of the unit noml Ns whee Ts Ns =. Tht is, T = dt ds = κn. 2 We cn esily deive one of the cuvtue definitions nd 2 fom the othe. Fo instnce, if we stt with 2, then κ = T N = dt ds N = lim s 0 = lim s 0 = lim s 0 = dφ ds. Ts + s Ts N s φ T s φ s Ts + s Ts Ts Ts + s Ns φ Exmple. Let us compute the cuvtue of the unit-speed cicle αs = cos s, sin s. We obtin tht T = α s = sin s, cos s, N = cos s, sin s, T = α s = cos s, sin s = N. 2

3 Thus κs =. cf. 2 The cuvtue of cicle equls the invese of its dius eveywhee. The next esult shows tht unit-speed plne cuve is essentilly detemined once we know its cuvtue t ech point of the cuve. The mening of essentilly hee is up to igid motion of R 2. Theoem Let κ :,b R be n integble function. Then thee exists unit-speed cuve α :,b R 2 whose cuvtue is κ. Futhemoe, if α :,b R 2 is ny othe unit-speed cuve with the sme cuvtue function κ, thee exists igid body motion tht tnsfoms α into α. Poof Fix s 0,b nd define, fo ny s,b, φs = αs = κudu, cf., s 0 s cos φtdt, sinφtdt. s 0 s 0 Then, the tngent vecto of α is α s = cos φs,sin φs, which is unit vecto mking n ngle φs with the x-xis. Thus α is unit speed, nd hs cuvtue dφ ds = d ds s 0 κudu = κs. Fo poof of the second pt, we efe to [3, p. 3]. The bove theoem shows tht we cn find plne cuve with ny given smooth function s its signed cuvtue. But simple cuvtue cn led to complicted cuves, s shown in the next exmple. Exmple 2. Let the signed cuvtue be κs = s. Following the poof of Theoem, nd tking s 0 = 0, we get φs = αs = u du = s2 0 2, cos s2 2 ds, 0 0 sin s2 2 ds. These integls cn only be evluted numeiclly. 2 The cuve is dwn in the figue below. 3 A igid motion consists of ottion nd tnsltion. 2 They ise in the theoy of diffction of light, whee they e clled Fesnel s integls, nd the cuve is clled Conu s Spil, lthough it ws fist consideed by Eule. 3 Tken fom [3, p. 33]. 3

4 When the cuvtue κs > 0, the cente of cuvtue lies long the diection of Ns t distnce κ fom the point αs. When κs < 0, the cente of cuvtue lies long the diection of Ns t distnce κ fom αs. In eithe cse, the cente of cuvtue is locted t αs + κs Ns. Hee, κ is clled the dius of cuvtue. The osculting cicle, when κ 0, is the cicle t the cente of cuvtue with dius κ. It ppoximtes the cuve loclly up to the second ode. osculting cicle cente of cuvtue dius of cuvtue /κ αs The totl cuvtue ove closed intevl [, b] mesues the ottion of the unit tngent Ts s s chnges fom to b: Φ, b = = = κds dφ ds ds dφ = φb φ. 4

5 If the totl cuvtue ove [,b] is within [0,2π], it hs closed fom: ccos T Tb, if T Tb 0; Φ,b = 2π ccos T Tb, othewise. When the tngent mkes sevel full evolutions 4 s s inceses fom to b, the totl cuvtue cnnot be detemined just fom T nd Tb. T α T totl cuvtue θ Tb Tb αb A point s on the cuve α is simple inflection, o n inflection, if the cuvtue κs = 0 but κ s 0. Intuitively, simple inflection is whee the cuve swing fom the left of the tngent t the point to its ight; o in the cse of simple closed cuve, it is whee the closed cuve α chnges fom convex to concve o fom concve to convex. In the figue below, the cuve on the left hs one simple inflection while the cuve on the ight hs six simple inflections. κ < 0 simple inflection κ < 0 κ < 0 κ < 0 In genel, point s with κs = κ s = = κ j s = 0 nd κ j s 0 is n inflection point of ode j. A second ode inflection point, lso efeed to s point of simple undultion, will not lte the convexity o concvity of its neighbohood on simple closed cuve. A simple vetex, o vetex, of cuve stisfies κ = 0 but κ 0. Intuitively, simplex vetex is whee the cuvtue ttins locl minimum o mximum. Fo exmple, n ellipse hs fou vetices, on its mjo nd mino xes. 4 Fo exmple, the cuve is the Conu s Spil. 5

6 simple vetex 2 Cuvtue of Abity-Speed Cuves Let αt be egul but not necessily unit-speed cuve. We obtin the unit tngent s T = α / α nd the unit noml N s the counteclockwise ottion of T by π 2. Still denote by κt the cuvtue function. Let αs be the unit-speed epmetiztion of α, whee s is n c-length function fo α. Let T = d α/ds be the unit tngent nd κs the cuvtue function unde this unit-speed pmetiztion. The cuvtue t point is independent of ny pmetiztion so κt = κst. Also by definition Tt = Ts. Diffeentite this eqution nd pply the chin ule: T t = T s ds dt. 3 Since αs is unit-speed, we know tht T s = κsñs. Substituting the function s in this eqution yields T s = κ stñ st = κtnt 4 by the definition of κ nd N in the bity-speed cse. We know tht ds/dt = α t fom the definition of c length s = t t 0 α u du. Denote by v = α t the speed function of α. Equtions 3 nd 4 combine to yield Now let αt = xt, yt. Then T = κvn. 5 T = x,y / α t = x,y / x 2 + y 2, N = y,x / x 2 + y 2. Substituting these tems into 5 yields fomul fo evluting the cuvtue: κ = T N v 6

7 = x,y x 2 + y + d x,y y,x / x 2 dt 2 + y 2 x 2 + y 2 x 2 + y 2 = x y x y. 6 x 2 + y We cn wite the fomul simply s κ = α α α 3, 7 by teting the coss poduct s scl. The denominto α 3 cn be egded s coection to diffeentitions of α when the cuve is not unit-speed: division by α once fo the velocity α, second time fo the noml, nd thid time fo the cceletion α. When the cuve is unit speed, α = T nd α = T = κn. The fomul 7 becomes n identity unde T N =. Exmple 3. Find the cuvtue of the cuve αt = t 3 t, t 2. so we hve Theefoe α t = 3t 2, 2t, α t = 6t, 2. κ = x y x y x 2 + y = 3t2 2 2t 6t 3t t t =. 9t 4 2t Finlly, let us deive the fomul fo the totl cuvtue ove [,b]. Let αs be the unit-speed pmetiztion of α, whee s is the c length function. Let ã nd b be the pmete vlues such tht αã = α nd α b = αb. Then the totl cuvtue of α ove [ã, b] is given by b ã κs ds. Since ds/dt = α t, we substitute t fo s in the bove eqution nd obtin the totl cuvtue fomul Refeences Φ,b = κt α t dt. [] B. O Neill. Elementy Diffeentil Geomety. Acdemic Pess, Inc., 966. [2] J. W. Rutte. Geomety of Cuves. Chpmn & Hll/CRC, [3] A. Pessley. Elementy Diffeentil Geomety. Spinge-Velg London,