( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that


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1 Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we hve lredy lerned for rc length: Definition. The length of the smooth curve trced out by from to b (where < b) is given by b (df ) ( ) dg L + + ( dh r(t) f(t) i+g(t) j +h(t) k s t vries ). Recll tht so tht its length is r (t) is given by r (t) r (t) df i + dg j + dh k, (df ) + ( dg ) + ( dh Compring this eqution with the one listed bove for the length of see tht b (df ) ( ) dg ( ) dh b + + ). r (t). r(t) from t to t b, we So n lternte (nd often more convenient) form for the length of the curve trced out by from to b is b L r (t). In summry, Arc Length of Curve The length of the smooth curve trced out by b (where < b) is given by b (df ) L + b r (t). r(t) r(t) f(t) i + g(t) j + h(t) k s t vries from to ( dg ) + ( dh ) Sometimes, we will choose prticulr point (clled the bse point) on the curve, nd mesure (directed) distnces long the curve using s the strting point. By directed distnce, we men tht the distnce will be positive number when we move in the direction of incresing t, nd negtive if we move in the direction of decresing t. For exmple, consider the spirl r(t) (cos t) i + (sin t) j + t k,
2 nd let. Then r() 0 i + j + k j + k. Then we my clculte the directed distnce long the curve from t to t, or from t to t 0. Since r (t) (sin t) i + (cos t) j + k, the directed distnce from to t is: r (t) t sin t + cos t We ve mesured the length of the purple portion of the curve below: Similrly, the directed distnce long the curve from t to t 0 is 0 r (t) t sin t + cos t
3 This number is the directed distnce long the curve from t to t 0, or the length of the green portion of the curve below: Choosing bse point ctully defines function which we cll s(t); set s(t) v(t). Then s(t) is the directed distnce long the curve trced out by r(t) from to t. Consider the exmple bove with. Since the distnce long the curve from t to t is., s().. Similrly, since the distnce long the curve from t to t 0 is ., s(0).. Definition Let r(τ) be vector function with derivtive t. Then the rc length prmeter is the function ( ) s(t) r df ( ) dg (τ) dτ + + dτ dτ we interpret s(t) s the length of the curve from to t. r (τ), nd choose prmeter ( dh dτ ) dτ; Note tht the chnge of function vrible from t to τ in r is only cosmetic, to prevent confusing the t in the integrl with t in the function. Writing r(τ) nd using τ is exctly the sme s writing r(t) if we were using t s the input for r; we switch from t to τ in r only so tht we my use the vrible t s the input for the function s without mbiguity. Exmple With r(t) (cos t) i + (sin t) j + t k nd t 0, the rc length prmeter is 3
4 s(t) τ r (τ) dτ sin τ + cos τ + dτ + dτ dτ t (t ). So the function s(t) is given by s(t) (t ). We cn rewrite the eqution bove in terms of s(t) s insted of t. Writing s (t ), solving for s, nd thinking of t s t(s), we hve t(s) s +. Curvture Recll from the previous section tht, given vector function r(t) nd the curve C tht it trces out s t vries, the vector r T (t) r (t) is the unit vector tngent to C t t. For ese of drwing, we will consider two dimensionl curves (where r(t) does not hve k component); however, the discussion is ctully pplicble to curves in three dimensionl spce s well. T cn be used to give us more informtion bout r(t) nd C; in prticulr, we cn use it to describe the wy tht C curves. Consider the two curves below: 4
5 The second one curves gret del more thn the first one does. Compring the unit tngent vectors for ech, we see tht the tngent vectors for the second curve chnge drmticlly in comprison with the those of the first; so with the rte of chnge of T in mind, we expect tht the derivtive of T is much greter in the second cse thn in the first. To mke the discussion precise, we define the curvture function: Definition If function of C is r(t) trces out the curve C nd hs unit tngent vector T, then the curvture κ ds. Since it is often difficult to determine the rc length prmeter s, it is convenient to rewrite the previous formul in terms of the vrible t. Thinking of t s function of s, we know by the chin rule tht dt ds d T ds. Then ds dt ds ds dt r (t). So we hve the following reformultion of the curvture formul: is Curvture of Smooth Curve If r(t) trces out the curve C nd hs unit tngent vector T, then the curvture function of C Alterntively, κ ds T (t) r (t). κ r (t) r (t). r (t) 3 Notice tht κ is sclr function it outputs numbers tht describe the curvture of C, not vectors. The lrger the vlue for κ, the more drmtic curve we expect to see. Principl Unit Norml Recll tht the function T is the unit tngent vector to the curve defined by r(t) in prticulr, T is lwys. So T is vector function of constnt length, which mens tht its derivtive d T ds is orthogonl to T. It will often be convenient to hve unit vector orthogonl to T ; we cn crete 5
6 this by dividing d T ds by its length. Fortuntely, we know tht d T κ. So we crete the following ds vector function: Definition Given curve C with unit tngent vector T, the principl unit norml vector for C is defined whenever κ 0 by N dt κ ds T (t) T (t). The principl unit norml vector N for C is the vector function so tht () (b) (c) N(t 0 ) is orthogonl to T (t 0 ) for ny t 0, N(t 0 ) is unit vector, nd N(t 0 ) points in the direction of the curvture of C. If we compute the cross product T N, we end up with third vector tht is orthogonl to both T nd N. We set B T N nd cll B the binorml vector of the curve r(t). The vectors B, T, nd N re mutully orthogonl; we cn think of them s giving us new xes tht re more relevnt to the curve thn re the x, y, nd z xes. Indeed, s the prmeter t in r(t) increses, we re provided with moving frme of reference for the curve: We cll this the TNB frme or the Frenet frme. 6
7 Figure : The red vectors re T s, blue vectors re Ns, nd the green vectors re Bs. The vectors T nd N define plne; vector B describes the tendency of the curve to rise orthogonlly out of this plne, nd will point in the direction in which r(t) is leving this plne. Exmple Given the curve r(t) ln(cos t) i + t j + k with < t <, find κ nd N. Determine the vlue for κ t t 0 nd t 3, nd interpret the results. Then find the principl norml vector t t 0. In order to fine κ, we ll need to clculte T. Since T r (t) sin t cos t i + j tn t i + j. r (t), we need to find r r (t). (t) 7
8 So r T (t) r (t) r (t) ( tn t i + j) tn t + ( tn t i + j) sec t ( tn t i + j) sec t ( tn t i + j) tn t sec t i + sec t j sin t i + cos t j. Since κ T (t), we need to know T r (t): (t) T (t) cos t i sin t j. So κ T (t) r (t) sec t cos t i sin t j cos sec t t + sin t sec t cos t. In order to clculte the unit norml vector, we will use the second formul for N: T N (t) T (t) cos t ( cos t i sin t j) i sin t j cos t + sin t ( cos t i sin t j) cos t i sin t j. 8
9 The curvture t t 0 is κ(0) cos 0 ; t t 3 the curvture is κ( 3 ) cos( 3 ). Since κ is lrger t t 0 thn t t 3, we expect tht the curve is more drstic t t 0, which is verified in the grph below of the curve: The principl unit norml vector t t 0 is N(0) cos 0 i sin 0 j i. Bounds on Curvture The curvture function κ v cn never be negtive, s the clcultion of the length of vector lwys produces nonnegtive number. However, if the vector function r(t) trces out line, we would expect the curvture to be 0. We cn esily show tht this is true. If r(t) ( 0 + t) i + (b 0 + bt) j + (c 0 + ct) k, where, 0, b, b 0, c, nd c 0 re constnts,then v(t) i + b j + c k, with v(t) + b + c. So we see tht T thus T is constnt, nd d T + b + c i + 0. In this cse, b + b + c j + κ v 0. c + b + c k; On the other hnd, we cn lso show tht there is no upper bound on κ; i.e., we cn mke κ s lrge s we like by choosing the pproprite curve. For exmple, it is known tht circle of rdius hs constnt curvture κ /. By decresing the circle s rdius, we my mke κ s lrge s we like; indeed, lim 0. In conclusion, we hve 0 κ <. 9
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