By Luther Pfahler Eisenhart
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Additional info for A Treatise on the Differential Geometry of Curves and Surfaces (Dover Phoenix Editions)
5, in which the curve is the are the correspond locus of the points M\ the points (7, C^ C2 are normal ing centers of curvature the planes MCN, M^C^N^ and the are the polar lines the lines CP, C^P^ to the curve This result is , ; ; ; points P, Pj, P 2 , are the centers of the osculating spheres. BEETEAND CUKVES 19. Bertrand proposed the following problem curves whose principal normals are the principal Bertrand curves. To determine the : In solving this problem we make use of must find the necessary and sufficient normals of another curve.
All o f From tbe definition of w it which pass through follows that w is C.. constant only when the curve C we have plane. j n this case we may take w equal to zero. Then when c the evol\te C in the plane of the curve. The other evolutes lie upon the right is MINIMAL CUEVES 47 cylinder formed of the normals to the plane at points of Co, and cut the elements of the cylinder under the constant angle 00 c, and consequently are helices. Hence we have the theorem : The evolutes of a plane curve are the helices traced on the right cylinder whose base is the plane evolute.
Our problem reduces then to the integration of And equation (65). Riccati equations. 14. Equation (65) ^=L+ (66) 2 may be written MO + NP, N are functions of s. This equation is a generalized where L, M, form of an equation first studied by Riccati, f and consequently is named for him. theory of curves As and Riccati equations occur frequently in the surfaces, we shall establish several of their properties. When a particular integral of a Riccati equation known, the general integral can be obtained by two quadratures.
A Treatise on the Differential Geometry of Curves and Surfaces (Dover Phoenix Editions) by Luther Pfahler Eisenhart