By Luther Pfahler Eisenhart

**Read Online or Download A Treatise on the Differential Geometry of Curves and Surfaces (Dover Phoenix Editions) PDF**

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**Additional resources for A Treatise on the Differential Geometry of Curves and Surfaces (Dover Phoenix Editions)**

**Sample text**

Represented in fig. 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.

P -i Hence, by Maclauriri pressed in the form - y") r and similar expressions s = (37) are substituted in obtain = ---- (y p From we (41) and #, y, z can be ex -i vv = Ct 2p J b p z s 2 s -f- 3 -f---, , 6 pr where p and r are the radii of first and second curvature at the = 0, and the unwritten terms are of the fourth and higher point s powers in From s. * Furthermore, the curve a point, when a point moves along a curve in the positive direction, it side of the osculating passes from the positive to the negative at a point, or vice versa, according as the torsion at the I/T at .

For the curve x (61) I y= ads, I fids, z=* I yd*, a, /3, 7 are the direction-cosines of the tangent, and dx 2 + dy* + dz 2 s measures the arc of the curve. x_l_ ds ds*~p 2 p* we get d^z^n. df~~p 2 /d*x\* /^>\ 2 /^\ = 1 W/ W/ W/ p* Hence if p be positive for all values of s, it is the radius of curva ture of the curve (61), and Z, m, n are the direction-cosines of the principal normal in the positive sense. , v are the direction-cosines of the binomial; hence and the third of (57) it follows that r is the radius of Therefore we have the following theorem in the theory of curves fundamental torsion of the curve.