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