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By Nikolaj N. Yanenko, M. Holt

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16) Thus κ2[β] = = (α ◦ h)h 2 + (α ◦ h)h (α ◦ h)h h3 |h |3 3 · J(α ◦ h) h (α ◦ h) · J (α ◦ h) (α ◦ h) 3 . 15). 21. Let β be a unit-speed curve in the plane. 17) β = κ2[β]J β . Proof. Differentiating β · β = 1, we obtain β · β = 0. Thus β must be a multiple of J β . 17) . Finally, we give simple characterizations of straight lines and circles by means of curvature. 22. Let α: (a, b) → R2 be a regular curve. (i) α is part of a straight line if and only if κ2[α](t) ≡ 0. (ii) α is part of a circle of radius r > 0 if and only if κ2[α] (t) ≡ 1/r.

46 CHAPTER 2. 5), we consider a portion pq of the cable between the lowest point p and an arbitrary point q. Three forces act on the cable: the weight of the portion pq, as well as the tensions T and U at p and q. If w is the linear density and s is the length of pq, then the weight of the portion pq is ws. 11. 6) |T| = |U| cos θ and w s = |U| sin θ. Let q = (x, y), where x and y are functions of s. 6) we obtain dy ws = tan θ = . 8) ds = dx 1+ dy dx 2 . 5) with a = ω/|T |. 5. CISSOID OF DIOCLES 47 Although at first glance the catenary looks like a parabola, it is in fact the graph of the hyperbolic cosine.

Ellipses These are perhaps the next simplest curves after the circle. The name ‘ellipse’ (which means ‘falling short’) is due to Apollonius4 . 26) ellipse[a, b](t) = a cos t, b sin t , 0 t < 2π. 8: An ellipse and its foci The curvature of this ellipse is κ2[ellipse[a, b]](t) = (b2 cos2 ab . t + a2 sin2 t)3/2 4 Apollonius of Perga (262–180 BC). His eight volume treatise on conic sections is the standard ancient source of information about ellipses, hyperbolas and parabolas. 22 CHAPTER 1. 8) is illustrated below with an exaggerated vertical scale.

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