Download e-book for iPad: A Treatise on the Differential Geometry of Curves and by Luther Pfahler Eisenhart

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By Luther Pfahler Eisenhart

ISBN-10: 0486438201

ISBN-13: 9780486438207

Created specifically for graduate scholars, this introductory treatise on differential geometry has been a hugely winning textbook for a few years. Its strangely exact and urban process encompasses a thorough rationalization of the geometry of curves and surfaces, focusing on difficulties that would be so much precious to scholars. 1909 version.

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Extra resources for A Treatise on the Differential Geometry of Curves and Surfaces

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System of parallel curves on the tangent surface. When s is known the involutes are given directly (106). by equations Hence the complete de- termination of the involutes of a given curve requires one quadrature at most. From PIG. 9 the definition of t and above value, an involute can its be generated mechanically in the following manner, as represented 9. Take a fig. string of length c and bring it into coincidence with the curve, with one end at the point s = ; call the other in end A. former point be fixed and the string be unwound gradually from the curve beginning at A, this point will trace out an involute on the tangent surface.

2 Pv P ; , BEETRAND CURVES Bertrand proposed the following problem 19. Bertrand curves. To determine, moving : normals are the principal In solving this problem we make use of must find the necessary and sufficient the curves ivhose principal normals of another curve. the 39 trihedral. -axis of the moving trihedral. Since the point M^ remains on the moving Tj-axis, we condition that the point have d% 0, 77 Jc, 0) = df= 0. And since M^ tends to move at = 0. Now equations (82) reduce to Srj right angles to this axis, * fU-*, as p (96) From the second we *.

Hence, in the neighborhood of an ordinary point, the curve lies entirely on one side of the plane determined by the tangent and binormal on ciently small values of it is , the side of the positive direction of the principal normal. These properties of a twisted curve are discovered, likewise, from a consideration of the projections upon the coordinate planes of the approximate curve, whose equations consist of the first m The projection on the osculating plane is the (53). z = x s, parabola y = s /2 />, whose axis is the principal normal to the curve.

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

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