By Luther Pfahler Eisenhart
Created specially for graduate scholars, this introductory treatise on differential geometry has been a hugely winning textbook for a few years. Its strangely unique and urban strategy features a thorough clarification of the geometry of curves and surfaces, targeting difficulties that may be such a lot necessary to scholars. 1909 variation.
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Additional info for A Treatise on the Differential Geometry of Curves and Surfaces
R, we return to the consideration M does not have its limiting position, and we r denote respectively coordinates of the cenlet JT, T, Z\ x ter of the circle, the direction-cosines of the diameter through of the circle, when l lp m v 7^5 M M and the radius. If xv yv z l be the coordinates of v they have the is on the circle, we have values (17), and since l M If we notice that 2^ = divide through by e\ where i? 0, and after reducing the above equation we have involves terms of the limit r t becomes r, first and higher orders in 2a/71 becomes S^'7, that is e.
1? /> Eliminating and 77 and consequently of s, for the determination of (???.. = -f . 0, t> do- , we have Hence y can be found by quadratures as a function and then is given directly. of 0-, CURVES IN SPACE 36 Problem. Find a necessary and sufficient condition that a curve lie upon a sphere. denote the coordinates of the center, and E the radius of the sphere, we have | 2 + ip + f2 = JR 2 Since the center is fixed, the derivatives of , 17, are the resultgiven by (84). Consequently, when we differentiate the above equation, = 0, which shows that the normal plane to the curve ing equation reduces to If 17, , .
We have made As these imaginary curves are of interest in certain parts of the theory of surfaces, we devote this closing section to their discussion. The equation of condition may be i_ where v is written in the form /a' a constant or a function of u. equivalent to the following: These equations are CURVES IN SPACE 48 At most, the common ratio is a function of w, say /(ft). And so as they can be if we disregard additive constants of integration, can replace we in curve the a translation of space, removed by the above equations by x (109) We consider first the case when v is constant and call it a.
A Treatise on the Differential Geometry of Curves and Surfaces by Luther Pfahler Eisenhart