By Michael Spivak

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**Extra resources for A Comprehensive Introduction to Differential Geometry, VOL. 3, 2ND EDITION **

**Example text**

M find that the distance to second order at most. TJie osculating crossed ~by nearest to the From plane the curve, a twisted curve at an ordinary point to and of all the planes through the point is it lies curve. the second of (53) seen that y is positive for suffior positive negative. Hence, in the neighborhood of an ordinary point, the curve lies entirely on one side of the plane determined by the tangent and binormal on ciently small values of it is , the side of the positive direction of the principal normal.

P; hence the center of the sphere tiated, we get t\ = If this equation be differenon the polar line for each is Another differentiation gives, together with the preceding, the following coordinates of the center of the sphere point. : 1=0, (91) When the last of these equations (92) , is J Conversely, when = P> f = -Tp'. differentiated + we obtain the desired condition (rpO'=0. this condition is satisfied, the point with the coordinates (91) is lies and at constant distance from points of the curve.

Determine the order of the minimal curves for which the function /in (113) satisfies the condition 5. A (X + B* + C2 = 0. A plane whose equation is that the osculating plane of a minimal curve can be written B + (Z Show 4/'"/v __ 5/1*2 = o. that the equations of a minimal curve, for which /in (113) satisfies the 5/lv2 = c/'"8 , where a is a constant, can be put in the form condition 4/"'/ v x = 8 cos i, = 8- sin . y . z i, = 8*. t. GENERAL EXAMPLES 1. Show that the equations of any plane curve can be put in the form x = Jcos 0/(#) cfy, y and determine the geometrical significance of 2.