DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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12 Chapter 1. CurvesSummarizing,κ(s) = a c 2 =Now we deal with B and the torsion:aa 2 + b 2 and N(s) = ( − cos(s/c), − sin(s/c), 0 ) .B(s) =T(s) × N(s) = 1 c(b sin(s/c), −b cos(s/c),a)B ′ (s) = 1 c 2 (b cos(s/c),bsin(s/c), 0)= −bc 2 N(s),so we infer that τ(s) = bc 2 = ba 2 + b 2 .Note that both the curvature and the torsion are constants. The torsion is positive when thehelix is “right-handed” (b >0) and negative when the helix is “left-handed” (b
§2. Local Theory: Frenet Frame 13Using the more casual—but convenient—Leibniz notation for derivatives,dTdt = dTdTdsdT= υκN or κN =ds dt ds = dt= 1 ds υExample 2. Let’s calculate the curvature of the tractrix (see Example 2 in Section 1). Usingthe first parametrization, we have α ′ (θ) =(− sin θ + csc θ, cos θ), and sodtdTdt .υ(θ) =‖α ′ (θ)‖ = √ (− sin θ + csc θ) 2 + cos 2 θ = √ csc 2 θ − 1=− cot θ.(Note the negative sign because π 2≤ θ0 and (sin θ, − cos θ) isaunit vector we conclude thatκ(θ) =− tan θ and N(θ) =(sin θ, − cos θ).Later on we will see an interesting geometric consequence of the equality of the curvature and the(absolute value of) the slope. ▽Example 3. Let’s calculate the “Frenet apparatus” for the parametrized curveα(t) =(3t − t 3 , 3t 2 , 3t + t 3 ).We begin by calculating α ′ and determining the unit tangent vector T and speed υ:Nowα ′ (t) =3(1 − t 2 , 2t, 1+t 2 ),υ(t) =‖α ′ (t)‖ =3 √ (1 − t 2 ) 2 +(2t) 2 +(1+t 2 ) 2 =3 √ 2(1 + t 2 ) 2 =3 √ 2(1 + t 2 )T(t) = √ 1 12 1+t 2 (1 − t2 , 2t, 1+t 2 )= √ 1 ( )1 − t22 1+t 2 , 2t1+t 2 , 1 .κN = dT dTds = dtdsdtso= 1 dTυ(t) dt(1 −4t√2 (1 + t 2 ) 2 , 2(1 − )t2 )(1 + t 2 ) 2 , 01=3 √ 2(1 + t 2 )1=3 √ 1 2√ ·2(1 + t 2 ) 2 1+t 2} {{ }κ(−2t1+t 2 , 1 − )t21+t 2 , 0 .} {{ }NHere we have factored out the length of the derivative vector and left ourselves with a unit vectorin its direction, which must be the principal normal N; the magnitude that is left must be theand
Page 1: DIFFERENTIALGEOMETRY:A First Course
Page 6 and 7: 4 Chapter 1. Curves2πb2πaFigure 1
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Page 20 and 21: 18 Chapter 1. Curvesalso Exercise 2
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62 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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