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 particular and urban technique incorporates a thorough rationalization of the geometry of curves and surfaces, targeting difficulties that might be such a lot precious to scholars. 1909 variation.
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Additional resources for A Treatise on the Differential Geometry of Curves and Surfaces
When 13. curve is all the osculating planes of a curve pass through a fixed point, the plane. 14. Determine f(u) so that the curve What is the form of the curve ? x = a cos it, y = a sin z it, =/(w) shall be plane. 13. Intrinsic equations. two curves defined Fundamental theorem. Let C^ and C2 be s, and let points in terms of their respective arcs We upon each with the same values of s correspond. assume, furthermore, that at corresponding points the radii of first curvature have the same value, and also the radii of second curvature.
Illustrative examples. As an example of the foregoing method we consider is the locus of a point on the tangent to a twisted curve C at a constant distance a from the point of contact. the curve which The coordinates are a, 0, 0. In of the point MI of the curve with reference to the axes at this case equations (82) reduce to M CUBYES IN SPACE 84 Hence if from the point corresponding s x denotes the length of arc of C'i to s = on C, we have and the direction-cosines of the tangent to C\ with reference to the moving axes are given by .
Intrinsic equations are S that the helix whose intrinsic equations are p a cylinder whose cross-section is a catenary. = r = (s2 + 4)/V5 lies Establish the following properties for the curve with the intrinsic equations = Is, where a and 6 are constants as, T p (a) the Cartesian coordinates are reducible to x=Aeht cos y=Aeht sm i, zBeht , ^ where A, B, h are functions of a and 6; lies upon a circular cone whose axis coincides with the z-axis and (6) the curve cuts the elements of the cone under constant angle.
A Treatise on the Differential Geometry of Curves and Surfaces by Luther Pfahler Eisenhart