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By Michael Spivak

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Additional info for A Comprehensive Introduction to Differential Geometry, Vol. 3, 3rd Edition

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Darboux and E. " (Compare G. Darboux, Legons I; E. ) CONTACT 1-71 23 19. Spherical image. When we move all unit tangent vectors t of a curve C to a point, their end points will describe a curve on the unit sphere, called the spherical image (spherical indicatrix) of C. Show that the absolute value of the curvature is equal to the ratio of the arc length dst of the spherical image and the arc length of the curve ds. What is the spherical image (a) of a straight line; (b) of a plane curve; (c) of a circular helix?

1 CURVES 30 exists a unique set of continuous solutions of this system which assumes given values y°, yz..... * We deduce from this theorem that we can find in one and in only one way three continuous solutions ai(s), 01(s), yi(s) which assume for s = so the values 1, 0, 0 respectively. We can similarly find three continuous solutions a2, 02, 72, so that a2(so) = 0, 'Y2(50) = 0, l32(so) = 1, and three more continuous solutions a3, 03, 73, so that a3(80) = 0, 03(60) = 0, 73(50) = 1. The Eqs. (8-6) lead to the following relations between the a, 0,,y: 2d (a2 + F+1 + 71) = Kola, - ICalo1 + r7'iF3l - TYlyl = 0, or + 01 + 71 = const = 1 + 0 + 0 = 1.

Then, developing the involute from P via B, A, C back to P, we obtain a closed curve with two tangents OVALS 1-13] 51 in every direction; the distance between two opposite points Q, R is QR = PQ + PR = arc PB + DB + DB + arc AB - are AC + arc CP = arc BC - F are AB - are AC + 2DB. The curvature vector of the involute, which at Q is along QP, always points to the interior. Curves of constant width were introduced by L. Euler, De curves triangularibus, Acta Acad. Petropol. 1778 (1780) II, pp. 3-30, who called them orbiform curves, and the curve ABC of Fig.

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