DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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18 Chapter 1. Curvesalso Exercise 23); the rectifying (“straightening”) plane is the one that comes closest to flatteningthe curve near P ; the normal plane is normal (perpendicular) to the curve at P . (Cf. Figure 1.3.)EXERCISES 1.21. Compute the curvature of the following arclength-parametrized curves:a. α(s) = ( √2 1 1cos s, √2 cos s, sin s )b. α(s) = (√ 1+s 2 , ln(s + √ 1+s 2 ) )*c. α(s) = ( 13 (1 + s)3/2 , 1 3 (1 − s)3/2 1, √2 s ) , s ∈ (−1, 1)2. Calculate the unit tangent vector, principal normal, and curvature of the following curves:a. a circle of radius a: α(t) =(a cos t, a sin t)b. α(t) =(t, cosh t)c. α(t) =(cos 3 t, sin 3 t), t ∈ (0,π/2)3. Calculate the Frenet apparatus (T, κ, N, B, and τ) ofthe following curves:*a. α(s) = ( 13 (1 + s)3/2 , 1 3 (1 − s)3/2 1, √2 s ) , s ∈ (−1, 1)b. α(t) = ( 12 et (sin t + cos t), 1 2 et (sin t − cos t),e t)*c. α(t) = (√ 1+t 2 ,t,ln(t + √ 1+t 2 ) )d. α(t) =(e t cos t, e t sin t, e t )e. α(t) =(cosh t, sinh t, t)♯ 4. Prove that the curvature of the plane curve y = f(x) isgiven by κ =♯ *5.|f ′′ |(1 + f ′2 ) 3/2 .Use Proposition 2.2 and the second parametrization of the tractrix given in Example 2 ofSection 1 to recompute the curvature.*6. By differentiating the equation B = T × N, derive the equation B ′ = −τN.♯ 7. Suppose α is an arclength-parametrized space curve with the property that ‖α(s)‖ ≤‖α(s 0 )‖ =R for all s sufficiently close to s 0 . Prove that κ(s 0 ) ≥ 1/R. (Hint: Consider the functionf(s) =‖α(s)‖ 2 . What do you know about f ′′ (s 0 )?)8. Let α be a regular (arclength-parametrized) curve with nonzero curvature. The normal line toα at α(s) isthe line through α(s) with direction vector N(s). Suppose all the normal lines toα pass through a fixed point. What can you say about the curve?9. a. Prove that if all the normal planes of a curve pass through a particular point, then thecurve lies on a sphere. (Hint: Apply Lemma 2.1.)*b. Prove that if all the osculating planes of a curve pass through a particular point, then thecurve is planar.10. Prove that if κ = κ 0 and τ = τ 0 are nonzero constants, then the curve is a (right) circular helix.(Hint: The only solutions of the differential equation y ′′ +k 2 y =0are y = c 1 cos(kt)+c 2 sin(kt).)
§2. Local Theory: Frenet Frame 19Remark. It is an amusing exercise to give a and b (in our formula for the circular helix)in terms of κ 0 and τ 0 .*11. Proceed as in the derivation of Proposition 2.2 to show thatτ = α′ · (α ′′ × α ′′′ )‖α ′ × α ′′ ‖ 2 .12. Let α be a C 4 arclength-parametrized curve with κ ≠0. Prove that α is a generalized helix ifand only if α ′′ · (α ′′′ × α (iv) )=0. (Here α (iv) denotes the fourth derivative of α.)13. Suppose κτ ≠0atP .Ofall the planes containing the tangent line to α at P , show that α lieslocally on both sides only of the osculating plane.14. Let α be a regular curve with κ ≠0atP . Prove that the planar curve obtained by projectingα into its osculating plane at P has the same curvature at P as α.15. A closed, planar curve C is said to have constant breadth µ if the distance between paralleltangent lines to C is always µ. (No, C needn’t be a circle. See Figure 2.5.) Assume for therest of this problem that the curve is C 2 and κ ≠0.(the Wankel engine design)Figure 2.5a. Let’s call two points with parallel tangent lines opposite. Prove that if C has constantbreadth µ, then the chord joining opposite points is normal to the curve at both points.(Hint: If β(s) isopposite α(s), then β(s) =α(s)+λ(s)T(s)+µN(s). First explain whythe coefficient of N is µ; then show that λ = 0.)b. Prove that the sum of the reciprocals of the curvature at opposite points is equal to µ.(Warning: If α is arclength-parametrized, β is quite unlikely to be.)16. Let α and β be two regular curves defined on [a, b]. We say β is an involute of α if, for eacht ∈ [a, b],(i) β(t) lies on the tangent line to α at α(t), and(ii) the tangent vectors to α and β at α(t) and β(t), respectively, are perpendicular.Reciprocally, we also refer to α as an evolute of β.a. Suppose α is arclength-parametrized. Show that β is an involute of α if and only ifβ(s) =α(s)+(c − s)T(s) for some constant c (here T(s) =α ′ (s)). We will normally refer
Page 1: DIFFERENTIALGEOMETRY:A First Course
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68 Chapter 2. Surfaces: Local Theor
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76 Chapter 3. Surfaces: Further Top
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100 Chapter 3. Surfaces: Further To
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108 Appendix. Review of Linear Alge
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ANSWERS TO SELECTED EXERCISES1.1.1
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116 SELECTED ANSWERS2.4.8 Only circ
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118 INDEXisometry, 107K, 48k-point
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DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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