Differential Geometry

A Treatise on the Differential Geometry of Curves and by Luther Pfahler Eisenhart

By Luther Pfahler Eisenhart

Created specially for graduate scholars, this introductory treatise on differential geometry has been a hugely profitable textbook for a few years. Its strangely specific and urban method contains a thorough rationalization of the geometry of curves and surfaces, targeting difficulties that might be such a lot important to scholars. 1909 variation.

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Derive the following properties of the twisted cubic straight line is In any plane there planes can be drawn. i^(a) the only real curve of zero curvature at every point. is one line, : and only one, through which two osculating \f (b) Four fixed osculating planes are cut by the line of intersection of any two osculating planes in four points whose cross-ratio is constant. of the curve (c) Four planes through, a variable tangent and four fixed points are in constant cross-ratio. (d) What is the dual of (c) by the results of 7?

0i This reduces, in consequence of (76), to *= (79) One 63 = 0. for a and 6 we put as = oo, is given by taking oo and be substituted in (72), we get a 4- b = 0, where i = 1, 2. So becomes &A- = 0f, where i = 1, 2. The solutions of this equation solution of this ; If these values that equation (79) are 61 = 0i, 6 2 = i0i. t t From (77) P = e0 Q = - 0i, 3, J? = e 8 - - 1, 1 ', so that 2vcM-l When , ft 7 the foregoing values are substituted in (73), in (61), we get E= (80) c Cco&tds, 2 From J and the resulting values of y= the last of these expressions we find that the tangent to the curve makes the direction of the elements of the cylinder.

If J/ be the point M on C it lies in the normal plane to C on CQ corresponding to its coordinates are of the form and Jf, consequently f , at where p and q are the distances from Jf to the binormal and principal normal respectively. These quantities p and q must be such that the line is, MM Q is tangent to the locus of J/ at this point, that we must have where tc denotes a factor of proportionality. values are substituted in these equations, we When the above get and two other equations obtained by replacing #, Z, X by ft, m, p and y, w, v Hence the expressions in parentheses vanish.

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