$2.LOCAL THEORY:FRENET FRAME 19 12.Let be a arclength-parametrized curve with0.Prove that is a generalized helix if and only if a" .(a .(Here denotes the fourth derivative of) 13.Supposeat P.Of all the planes containing the tangent line to a at P,show that a lies locally on both sides only of the osculating plane. 14.Let a be a regular curve with0at P.Prove that the planar curve obtained by projecting into its osculating plane at P has the same curvature at P as o. 15.A closed,planar curve C is said to have constan breadh if the distance between parallel tangent lines to C is always u.(No,C needn't be a circle.See Figure 2.5.)Assume for the rest of this problem that the curve is e2 and0. (the Wankel engine design) a.Let's call two points with parallel tangent lines opposite.Prove that if C has constant breadth then the chord joining opposite points is normal to the curve at both points.(Hint:If B(s)is opposite (s).then B(s)=(s)+(s)T(s)+uN(s).First explain why the coefficient of N is u; then show that入=0.) b.Prove that the sum of the reciprocals of the cu at opposite points is equal to(Warning:If is arclengh-parametrized,to be) 16.Let a and B be two regular curves defined on [a.b].We say B is an imolute of a if,for eacht [a.b]. (i)B()lies on the tangent line(),and ()the tangent)nd )perpendicula Reciprocally,we also refer to as an evolute of B. a.Suppose a is arclength-parametrized.Show that B is an involute of a if and only if B(s)= (s)+(c-s)T(s)for some constantc(here T(s)=a'(s)).We will normally refer to the curve B obtained with c=0as the involute of a.If you were to wrap a string around the curve starting ats=0.the involute is the path the end of the string follows as you unwrap it,always pulling the string taut,as illustrated in the case of a circle in Figure2.6. b.Show that the involute of a helix is a plane curve. c.Show that the involute of a catenary is a tractrix.(Hint:You do not need an arclength parametriza- tionl) d.Ifa is an arclength-parametrized plane curve,prove that the curve B given by B(5)=a(s)+-
÷2. LOCAL THEORY: FRENET FRAME 19 12. Let ˛ be a C 4 arclength-parametrized curve with ¤ 0. Prove that ˛ is a generalized helix if and only if ˛ 00 .˛ 000 ˛ .iv/ / D 0. (Here ˛ .iv/ denotes the fourth derivative of ˛.) 13. Suppose ¤ 0 at P. Of all the planes containing the tangent line to ˛ at P, show that ˛ lies locally on both sides only of the osculating plane. 14. Let ˛ be a regular curve with ¤ 0 at P. Prove that the planar curve obtained by projecting ˛ into its osculating plane at P has the same curvature at P as ˛. 15. A closed, planar curve C is said to have constant breadth if the distance between parallel tangent lines to C is always . (No, C needn’t be a circle. See Figure 2.5.) Assume for the rest of this problem that the curve is C 2 and ¤ 0. (the Wankel engine design) FIGURE 2.5 a. Let’s call two points with parallel tangent lines opposite. Prove that if C has constant breadth , then the chord joining opposite points is normal to the curve at both points. (Hint: If ˇ.s/ is opposite ˛.s/, then ˇ.s/ D ˛.s/ C .s/T.s/ C N.s/. First explain why the coefficient of N is ; then show that D 0.) b. Prove that the sum of the reciprocals of the curvature at opposite points is equal to . (Warning: If ˛ is arclength-parametrized, ˇ is quite unlikely to be.) 16. Let ˛ and ˇ be two regular curves defined on Œa; b. We say ˇ is an involute of ˛ if, for each t 2 Œa; b, (i) ˇ.t / lies on the tangent line to ˛ at ˛.t /, and (ii) the tangent vectors to ˛ and ˇ at ˛.t / and ˇ.t /, respectively, are perpendicular. Reciprocally, we also refer to ˛ as an evolute of ˇ. a. Suppose ˛ is arclength-parametrized. Show that ˇ is an involute of ˛ if and only if ˇ.s/ D ˛.s/ C .c s/T.s/ for some constant c (here T.s/ D ˛ 0 .s/). We will normally refer to the curve ˇ obtained with c D 0 as the involute of ˛. If you were to wrap a string around the curve ˛, starting at s D 0, the involute is the path the end of the string follows as you unwrap it, always pulling the string taut, as illustrated in the case of a circle in Figure 2.6. b. Show that the involute of a helix is a plane curve. c. Show that the involute of a catenary is a tractrix. (Hint: You do not need an arclength parametrization!) d. If ˛ is an arclength-parametrized plane curve, prove that the curve ˇ given by ˇ.s/ D ˛.s/ C 1 .s/N.s/
CHAPTER 1.CURVES FIGURE 2.6 is the unique evolute of lying in the plane of.Prove,moreover,that this curve is regular if .(Hint:Go back to the original definition.) l7.Find the involute of the cycloid(t)=(t+sint,1-cost),t∈【-π，πl,usingt=0 as your starting point.Give a geometric description of your answer. 18.Let be a curve parametrized by arclength with. a.Suppose lies on the surface of a sphere centered at the origin(i.e.l(s)=const for all s). Prove that (*) +(()=o (Hint:Write=T+N+vB for some functions,H,and v,differentiate,and use the fac that T.N.B is a basis for R') b.Prove the converse:If satisfies the differential equation ()then a lies on the surface of some sphere.(Hint:Using the values obtained in part a.show) is a constant vector,the candidate for the center of the sphere.) 19.Two distinct parametrized curves a and B are called Bertrand mates if for eacht,the normal line to a at()equals the normal line to Bat B().An example is pictured in Figure 2.7.Suppose and B are FIGURE 2.7 Bertrand mates. a.If a is arclength-parametrized,show that B(s)=a(s)+r(s)N(s)and r(s)=const.Thus. corresponding points ofand are a constant distance apart
20 CHAPTER 1. CURVES P FIGURE 2.6 is the unique evolute of ˛ lying in the plane of ˛. Prove, moreover, that this curve is regular if 0 ¤ 0. (Hint: Go back to the original definition.) 17. Find the involute of the cycloid ˛.t / D .t C sin t; 1 cost /, t 2 Œ; , using t D 0 as your starting point. Give a geometric description of your answer. 18. Let ˛ be a curve parametrized by arclength with ; ¤ 0. a. Suppose ˛ lies on the surface of a sphere centered at the origin (i.e., k˛.s/k D const for all s). Prove that (?) C 1 1 00 D 0: (Hint: Write ˛ D T C N C B for some functions , , and , differentiate, and use the fact that fT; N;Bg is a basis for R 3 .) b. Prove the converse: If ˛ satisfies the differential equation (?), then ˛ lies on the surface of some sphere. (Hint: Using the values of , , and you obtained in part a, show that ˛.TCNCB/ is a constant vector, the candidate for the center of the sphere.) 19. Two distinct parametrized curves ˛ and ˇ are called Bertrand mates if for each t, the normal line to ˛ at ˛.t / equals the normal line to ˇ at ˇ.t /. An example is pictured in Figure 2.7. Suppose ˛ and ˇ are FIGURE 2.7 Bertrand mates. a. If ˛ is arclength-parametrized, show that ˇ.s/ D ˛.s/ C r.s/N.s/ and r.s/ D const. Thus, corresponding points of ˛ and ˇ are a constant distance apart
$2.LOCAL THEORY:FRENET FRAME b.Show that,moreover,the angle between the tangent vectors to and B at corresponding points is constant.(Hint:If T and TB are the unit tangent vectors to and B respectively,consider Ta·Tg) c.Suppose is arclength-parametrized and.Show that has a Bertrand mate if and only if there are constants r and c so that rK+c=1. d.Given o.prove that if there is more than one curve 8 so that o and 8 are Bertrand mates.then there are infinitely many such curves B and this occurs if and only if o is a circular helix. 20.(See Exercise 19.)Suppose and B are Bertrand mates.Prove that the torsion of and the torsion of B at corresponding points have constant product. 21.Suppose Y is a e2 vector function on [a,b]with Yll =I and Y,Y,and Y"everywhere linearly independent.For any nonzero constantc,define()=c (Y(m)×Y'(w)du,t∈[a.bl.Prove that the curve a has constant torsion 1/c.(Hint:Show that B=+Y.) 22 a.Let a be an arclength-parametrized plane curve.We create a"parallel"curve B by taking B +eN(for a fixed small positive value of s).Explain the terminology and express the curvature of B in terms of e and the curvature of a. b.Now let be an arclength-parametrized sp ce curve.Show that we can obtain a"parallel"curve by taking =+((cos)N+(sin)B)for an appropriate function 0.How many such parallel curves are there? c Sketch such a parallel curve for a circular helix. 23.Suppose is an arclength-parametrized curve,P=(0),and K(0)0.Use Proposition 2.6 to establish the following: *a.Let =a(s)and R=(t).Show that the plane spanned by P,and R approaches the osculating plane of at Pass. b.The osculating circle at P is the limiting position of the circle passing through P,and R as s.t0.Prove that the osculating circle has center Z=P+(1/K(0))N(0)and radius 1/(0). c.The sphere at Pis the limiting position of the sphere through Pand three neighboring points on the curve,as the latter points tend to P independently.Prove that the osculating sphere has center Z=P+(1/k(O)N(O)+(1/x(O(1/k)/(O)B(O) and radius V(1/x(0)2+(1/x(0)(1/x)y(0)2 d.How is the result of part e related to Exercise 18? 24.a Suppose isaplane curve and C is the cirele at)wth radius()Assuming and rare differentiable functions,show that the circle Cs is contained inside the circle C whenever t>s if and only if lB'(s)r'(s)for all s. b.Let be arclength-parametrized plane curve and supposeis a decreasing function.Prove that the osculating circle at(s)lies inside the osculating circle at)whenevers.(See Exercise23 for the definition of the osculating circle.)
÷2. LOCAL THEORY: FRENET FRAME 21 b. Show that, moreover, the angle between the tangent vectors to ˛ and ˇ at corresponding points is constant. (Hint: If T˛ and Tˇ are the unit tangent vectors to ˛ and ˇ respectively, consider T˛ Tˇ.) c. Suppose ˛ is arclength-parametrized and ¤ 0. Show that ˛ has a Bertrand mate ˇ if and only if there are constants r and c so that r C c D 1. d. Given ˛, prove that if there is more than one curve ˇ so that ˛ and ˇ are Bertrand mates, then there are infinitely many such curves ˇ and this occurs if and only if ˛ is a circular helix. 20. (See Exercise 19.) Suppose ˛ and ˇ are Bertrand mates. Prove that the torsion of ˛ and the torsion of ˇ at corresponding points have constant product. 21. Suppose Y is a C 2 vector function on Œa; b with kYk D 1 and Y, Y 0 , and Y 00 everywhere linearly independent. For any nonzero constant c, define ˛.t / D c Z t a Y.u/ Y 0 .u/ du, t 2 Œa; b. Prove that the curve ˛ has constant torsion 1=c. (Hint: Show that B D ˙Y.) 22. a. Let ˛ be an arclength-parametrized plane curve. We create a “parallel” curve ˇ by taking ˇ D ˛ C "N (for a fixed small positive value of "). Explain the terminology and express the curvature of ˇ in terms of " and the curvature of ˛. b. Now let ˛ be an arclength-parametrized space curve. Show that we can obtain a “parallel” curve ˇ by taking ˇ D ˛ C " .cos /N C .sin /B for an appropriate function . How many such parallel curves are there? c. Sketch such a parallel curve for a circular helix ˛. 23. Suppose ˛ is an arclength-parametrized curve, P D ˛.0/, and .0/ ¤ 0. Use Proposition 2.6 to establish the following: *a. Let Q D ˛.s/ and R D ˛.t /. Show that the plane spanned by P, Q, and R approaches the osculating plane of ˛ at P as s; t ! 0. b. The osculating circle at P is the limiting position of the circle passing through P, Q, and R as s; t ! 0. Prove that the osculating circle has center Z D P C 1=.0/ N.0/ and radius 1=.0/. c. The osculating sphere at P is the limiting position of the sphere through P and three neighboring points on the curve, as the latter points tend to P independently. Prove that the osculating sphere has center Z D P C 1=.0/ N.0/ C 1= .0/.1=/0 .0/ B.0/ and radius q .1=.0//2 C .1= .0/.1=/0 .0//2: d. How is the result of part c related to Exercise 18? 24. a. Suppose ˇ is a plane curve and Cs is the circle centered at ˇ.s/ with radius r.s/. Assuming ˇ and r are differentiable functions, show that the circle Cs is contained inside the circle Ct whenever t > s if and only if kˇ 0 .s/k r 0 .s/ for all s. b. Let ˛ be arclength-parametrized plane curve and suppose is a decreasing function. Prove that the osculating circle at ˛.s/ lies inside the osculating circle at ˛.t / whenever t > s. (See Exercise 23 for the definition of the osculating circle.)
22 CHAPTER 1.CURVES 25.Suppose the front wheel of a bicycle follows the arclength-parametrized plane curve.Determine the path B of the rear wheel,I unit away,as shown in Figure 2.8.(Hint:If the front wheel is turned an FIGURE 2.8 angle from the axle of the bike,start by writing c-B in terms of T,and N.Your goal should be adifferential quation thatmust satisfy,invoing ony.Note that the path ofthe rear wheel will obviously depend on the initial condition().In all but the simplest of cases,it may be impossible to solve the differential equation explicitly.) 3.Some Global Results 3.1.Space Curves.The fundamental notion in geometry (see Section 1 of the Appendix)is that of congruence:When do two figures differ merely by a rigid motion?If the curveis obtained from the curve by performing a rigid motion(composition of a translation and a rotation),then the Frenet frames at corresponding points differ by that same rigid motion,and the twisting of the frames(which is what gives curvature and torsion)should be the same.(Note that a reflection will not affect the curvature,but will change the sign of the torsion.) Ther3.1 (Fundamental Theore of Curve Theory).Two space curves C and Care congru (by arigid motion)ifand only if the corresponding arclength parametrizations.L]R3 have the property that K(s)=K*(s)and r(s)=t*(s)for all s [0.L]. Proof.Suppose*=Ψfor some rigid motion平：R3→R3,soΨ(=A+b for some b∈ R3 and some 3 x 3 orthogonal matrix A with det A >0.Then (s)=Ac(s)+b,sol(s)= llAa'(s)l=1,since A is orthogonal.Therefore,a is likewise arclength-parametrized,and T*(s)= AT(s).Differentiating again,*(s)N*(s)=K(s)AN(s).Since A is orthogonal,AN(s)ll is a unit vector and so N*(s)=AN(s)and k*(s)=k(s).But then B*(s)=T*(s)x N*(s)=AT(s)x AN(s)= A(T(s)x N(s))=AB(s),inasmuch as orthogonal matrices map orthonormal bases to orthonormal bases and det A>0 insures that orientation is preserved as well.Last.B'(s)=-(s)N(s)and B'(s)= AB'(s)=-t(s)AN(s)=-7(s)N*(s).so *(s)=r(s).as required Conversely,suppose =k*and =*We now define a rigid motion as follows.Let b a*(0)-c(0),and let A be the unique orthogonal matrix so that AT(0)=T*(0),AN(0)=N*(0),and AB(0)=B*().A also has positive determinant,since both orthonormal bases are right-handed.Set
22 CHAPTER 1. CURVES 25. Suppose the front wheel of a bicycle follows the arclength-parametrized plane curve ˛. Determine the path ˇ of the rear wheel, 1 unit away, as shown in Figure 2.8. (Hint: If the front wheel is turned an FIGURE 2.8 angle from the axle of the bike, start by writing ˛ ˇ in terms of , T, and N. Your goal should be a differential equation that must satisfy, involving only . Note that the path of the rear wheel will obviously depend on the initial condition .0/. In all but the simplest of cases, it may be impossible to solve the differential equation explicitly.) 3. Some Global Results 3.1. Space Curves. The fundamental notion in geometry (see Section 1 of the Appendix) is that of congruence: When do two figures differ merely by a rigid motion? If the curve ˛ is obtained from the curve ˛ by performing a rigid motion (composition of a translation and a rotation), then the Frenet frames at corresponding points differ by that same rigid motion, and the twisting of the frames (which is what gives curvature and torsion) should be the same. (Note that a reflection will not affect the curvature, but will change the sign of the torsion.) Theorem 3.1 (Fundamental Theorem of Curve Theory). Two space curves C and C are congruent (i.e., differ by a rigid motion) if and only if the corresponding arclength parametrizations ˛; ˛ W Œ0; L ! R 3 have the property that .s/ D .s/ and .s/ D .s/ for all s 2 Œ0; L. Proof. Suppose ˛ D ‰ı˛ for some rigid motion ‰W R 3 ! R 3 , so ‰.x/ D Ax C b for some b 2 R 3 and some 3 3 orthogonal matrix A with det A > 0. Then ˛ .s/ D A˛.s/ C b, so k˛ 0.s/k D kA˛ 0 .s/k D 1, since A is orthogonal. Therefore, ˛ is likewise arclength-parametrized, and T .s/ D AT.s/. Differentiating again, .s/N .s/ D .s/AN.s/. Since A is orthogonal, kAN.s/k is a unit vector, and so N .s/ D AN.s/ and .s/ D .s/. But then B .s/ D T .s/ N .s/ D AT.s/ AN.s/ D A.T.s/ N.s// D AB.s/, inasmuch as orthogonal matrices map orthonormal bases to orthonormal bases and det A > 0 insures that orientation is preserved as well. Last, B 0.s/ D .s/N .s/ and B 0.s/ D AB 0 .s/ D .s/AN.s/ D .s/N .s/, so .s/ D .s/, as required. Conversely, suppose D and D . We now define a rigid motion ‰ as follows. Let b D ˛ .0/ ˛.0/, and let A be the unique orthogonal matrix so that AT.0/ D T .0/, AN.0/ D N .0/, and AB.0/ D B .0/. A also has positive determinant, since both orthonormal bases are right-handed. Set
$3.SOME GLOBAL RESULTS =o.We now claim that(s)=(s)for all s[L].Note,by our argument in the first part of the proof,that==*and==*.Consider fs)=Ts).T*s)+Ns)·N*(s)+B(s)·B*). We now differentiate f,using the Frenet formulas. f'()=(T'(s)·T*(s)+T(s)·T*'(s)+(N(s)N*(s)+N(s).N*(s) +(B'(s)·B*(s)+B(s)·B*'(s) =x(s)(S(s).T*(s)+(s).N*(s))-x(s)(i(s).N(s)+N(s).T*(s)) +ts)((s)Ns)+Ns)B*()-t)(N(s)·B*G)+B()·N*(s) =0, since the first two terms cancel and the last two terms cancel.By construction,f(0)=3,so f(s)=3 for allsL].Since each of the individual dot products can be at most 1,the only way the sum can be3 for all s is for each to be 1 for all s,and this in turn can happen only when T(s)=T(s).N(s)N(s).and B(s)=B*(s)for all s [0.L].In particular,since '(s)=T(s)=T*(s)=a(s)and (0)=a*(0).it follows that a(s)=a(s)for all s [0.L],as we wished to show. Remark The latter half of this proof can be replaced by asserting the uniqueness of solutions of a sys tem of differential equations,as we will see in amoment.Also see Exercise A.3.1 for a matrix-computational version of the proof we just did. Example 1.We now see that the only curves with constant and are circular helices. Perhaps more interesting is the existence question:Given continuous functionsK.:[0.L]R(withK everywhere positive),is there a space curve with those as its curvature and torsion?The answer is yes,and 3.I of the Appendix.That is,we let 「0-k)01 F(s)=T(s)N(s)B(s) and K(s)=k(d）0-t L0s)0 Then integrating the linear system of ordinary differential equations F(s)=F(s)K(s),F(0)=Fo.gives us the Frenet frame everywhere along the curve,and we recover by integrating T(s). We turn now to the concept of total curvature of a closed space curve,which is the integral of its curvature.That is.if [0.L]R3 is an arclength-parametrized curve with ()=(L).its total curvature is (s)ds.This quantity can be interpreted geometrically as follows:The Gauss map of is the map to the unit sphere,.Σ，given by the unit tangent vector T:0.L→Σ：its image,T,is classically called the tangen of Observe thatprovided the Gauss map isone-toonthe length ofis the total curvature of,since length(T)= )ds(sds.More generally,this integral is the length of T"counting multiplicities." A preliminary question to ask is this:What curves in the unit sphere can be the Gauss map of some closed space curve a?Since a(s)=a(0)+ T(u)du,we see that a necessary and sufficient condition
÷3. SOME GLOBAL RESULTS 23 ˛Q D ‰ı˛. We now claim that ˛ .s/ D Q˛.s/ for all s 2 Œ0; L. Note, by our argument in the first part of the proof, that Q D D and Q D D . Consider f .s/ D TQ.s/ T .s/ C NQ .s/ N .s/ C BQ.s/ B .s/: We now differentiate f , using the Frenet formulas. f 0 .s/ D TQ 0 .s/ T .s/ C TQ.s/ T 0.s/ C NQ 0 .s/ N .s/ C NQ.s/ N 0.s/ C BQ 0 .s/ B .s/ C BQ.s/ B 0.s/ D .s/ NQ .s/ T .s/ C TQ.s/ N .s/ .s/ TQ.s/ N .s/ C NQ .s/ T .s/ C .s/ BQ.s/ N .s/ C NQ.s/ B .s/ .s/ NQ .s/ B .s/ C BQ.s/ N .s/ D 0; since the first two terms cancel and the last two terms cancel. By construction, f .0/ D 3, so f .s/ D 3 for all s 2 Œ0; L. Since each of the individual dot products can be at most 1, the only way the sum can be 3 for all s is for each to be 1 for all s, and this in turn can happen only when TQ.s/ D T .s/, NQ .s/ D N .s/, and BQ.s/ D B .s/ for all s 2 Œ0; L. In particular, since ˛Q 0 .s/ D TQ.s/ D T .s/ D ˛ 0.s/ and ˛Q.0/ D ˛ .0/, it follows that ˛Q.s/ D ˛ .s/ for all s 2 Œ0; L, as we wished to show. Remark. The latter half of this proof can be replaced by asserting the uniqueness of solutions of a system of differential equations, as we will see in a moment. Also see Exercise A.3.1 for a matrix-computational version of the proof we just did. Example 1. We now see that the only curves with constant and are circular helices. O Perhaps more interesting is the existence question: Given continuous functions ; W Œ0; L ! R (with everywhere positive), is there a space curve with those as its curvature and torsion? The answer is yes, and this is an immediate consequence of the fundamental existence theorem for differential equations, Theorem 3.1 of the Appendix. That is, we let F .s/ D 2 6 4 j j j T.s/ N.s/ B.s/ j j j 3 7 5 and K.s/ D 2 6 4 0 .s/ 0 .s/ 0 .s/ 0 .s/ 0 3 7 5 : Then integrating the linear system of ordinary differential equations F 0 .s/ D F .s/K.s/, F .0/ D F0, gives us the Frenet frame everywhere along the curve, and we recover ˛ by integrating T.s/. We turn now to the concept of total curvature of a closed space curve, which is the integral of its curvature. That is, if ˛W Œ0; L ! R 3 is an arclength-parametrized curve with ˛.0/ D ˛.L/, its total curvature is Z L 0 .s/ds. This quantity can be interpreted geometrically as follows: The Gauss map of ˛ is the map to the unit sphere, †, given by the unit tangent vector TW Œ0; L ! †; its image, , is classically called the tangent indicatrix of ˛. Observe that—provided the Gauss map is one-to-one—the length of is the total curvature of ˛, since length./ D Z L 0 kT 0 .s/kds D Z L 0 .s/ds. More generally, this integral is the length of “counting multiplicities.” A preliminary question to ask is this: What curves in the unit sphere can be the Gauss map of some closed space curve ˛? Since ˛.s/ D ˛.0/ C Z s 0 T.u/du, we see that a necessary and sufficient condition