By Luther Pfahler Eisenhart
Created particularly for graduate scholars, this introductory treatise on differential geometry has been a hugely winning textbook for a few years. Its surprisingly designated and urban technique encompasses a thorough clarification of the geometry of curves and surfaces, focusing on difficulties that would be so much valuable to scholars. 1909 version.
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Additional info for A Treatise on the Differential Geometry of Curves and Surfaces
This result is true whatever be the relation (104)* Hence equation (105) gives the element of length of any curve on the surface, and da- is called the linear element of the surface. According as value, the point t in equations (102) has a positive or negative on the portion of the tangent drawn in the lies TANGENT SUBFACE OF A CURVE 43 positive direction from the curve or in the opposite direction. It is now our purpose to get an idea of the form of the surface in the neighborhood of the curve.
7. 8. Derive the results of is cubic. a curve whose binormals are the binormals of a 21 by means of the moving trihedral. Minimal curves. In the preceding discussion exception of th^ curves, defined by 22. =<>. 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).
From these equations it follows that a necessary and sufficient condition that this rate of change at a point be zero is that the values of s for the point make the At such determinant in equation (51) vanish. ing plane Form 11. of curve in the neighborhood of a point. We have torsion. a point the osculat- said to be stationary. is made The sign of the convention that the positive directions of the tangent, principal normal, and binormal shall have the same we relative orientation as the fixed x-} y-, s-axes respectively.
A Treatise on the Differential Geometry of Curves and Surfaces by Luther Pfahler Eisenhart