DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

22 Chapter 1. Curvesthe osculating sphere has center Z = P + ( 1/κ(0) ) N(0)+ ( 1/τ(0)(1/κ) ′ (0) ) B(0) and radius√(1/κ(0)) 2 +(1/τ(0)(1/κ) ′ (0)) 2 .d. How is the result of part c related to Exercise 18?24. a. Suppose β is a plane curve and C s is the circle centered at β(s) with radius r(s). Assumingβ and r are differentiable functions, show that the circle C s is contained inside the circleC t whenever t>sif and only if ‖β ′ (s)‖ ≤r ′ (s) for all s.b. Let α be arclength-parametrized plane curve and suppose κ is a decreasing function. Provethat the osculating circle at α(s) lies inside the osculating circle at α(t) whenever t>s.(See Exercise 23 for the definition of the osculating circle.)25. Suppose the front wheel of a bicycle follows the arclength-parametrized plane curve α. Determinethe path β of the rear wheel, 1 unit away, as shown in Figure 2.8. (Hint: If the frontFigure 2.8wheel is turned an angle θ from the axle of the bike, start by writing α − β in terms of θ, T,and N. Your goal should be a differential equation that θ must satisfy. Note that the path ofthe rear wheel will obviously depend on the initial condition θ(0). In all but the simplest ofcases, it may be impossible to solve the differential equation explicitly.)3. Some Global Results3.1. Space Curves. The fundamental notion in geometry (see Section 1 of the Appendix) isthat of congruence: When do two figures differ merely by a rigid motion? If the curve α ∗ is obtainedfrom the curve α by performing a rigid motion (composition of a translation and a rotation), thenthe Frenet frames at corresponding points differ by that same rigid motion, and the twisting of theframes (which is what gives curvature and torsion) should be the same. (Note that a reflection willnot 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 parametrizationsα, α ∗ :[0,L] → R 3 have the property that κ(s) =κ ∗ (s) and τ(s) =τ ∗ (s) for all s ∈ [0,L].Proof. Suppose α ∗ =Ψ◦α for some rigid motion Ψ: R 3 → R 3 ,soΨ(x) =Ax + b for someb ∈ R 3 and some 3 × 3 orthogonal matrix A with det A > 0. Then α ∗ (s) = Aα(s) +b, so‖α ∗′ (s)‖ = ‖Aα ′ (s)‖ =1,since A is orthogonal. Therefore, α ∗ is likewise arclength-parametrized,