A Treatise on the Differential Geometry of Curves and - download pdf or read online
By Luther Pfahler Eisenhart
Created particularly for graduate scholars, this introductory treatise on differential geometry has been a hugely profitable textbook for a few years. Its strangely targeted and urban method features a thorough clarification of the geometry of curves and surfaces, targeting difficulties that would be such a lot worthwhile to scholars. 1909 version.
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Extra info for A Treatise on the Differential Geometry of Curves and Surfaces
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? CUBVES IN SPACE 16 5. so that the principal normals to the Determine the form of the function cune x = = sin =0(it) are parallel to the 7/z-plane. 6. Find the osculating plane and radius of first curvature of M, y x = z if, a cos u + b sin w, y = a sin u + 6 cos w, It has 10. Torsion. Frenet-Serret formulas. = c sin 2 2 tt. been seen that, un- curve be plane, the osculating plane varies as the point moves along the curve.
An We shall When it apply the preceding results to several problems. is plane the torsion is zero, and conversely. For this case equa- the curve tion (65) reduces to = - as a 0, of which the general integral 2 J P = ae= ae-if where a is an arbitrary constant, and by spherical indicatrix of the tangent. (27) Particular integrals are the roots of the equation 2 -f 2 00 - 1 roots are real and unequal if c is real ; we consider only this case, and put = Two From (69) it follows that the general solution of the 6 (77) ~ above equation These is ' ofi*-l where we have put t (78) = e*~ Since
A Treatise on the Differential Geometry of Curves and Surfaces by Luther Pfahler Eisenhart
Particular integrals are the roots of the equation 2 -f 2 00 - 1 roots are real and unequal if c is real ; we consider only this case, and put = Two From (69) it follows that the general solution of the 6 (77) ~ above equation These is ' ofi*-l where we have put t (78) = e*~ Since