By Luther Pfahler Eisenhart
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Extra info for A Treatise on the Differential Geometry of Curves and Surfaces (Dover Phoenix Editions)
Lemons sur la Thdorie Generate des Surfaces, Vol. I, p. 22. We shall refer to this and for brevity give our references the form Darboux, I, 22. Cf Forsyth, Differential Equations, chap, v also Cohen, Differential Equations, treatise frequently, t is . pp. 173-177. ; CURVES IN SPACE 26 Let O l be a particular integral of (66). If is the equation for the determination of we put = l/c -f 0^ </> + 2(M+2WJ<l>+N**Q. - (67) As this equation is linear and of the first order, it can be solved by Since the general integral of (67) is of the form where a denotes the constant of integration, the two quadratures.
F This . is is given by of means by expressible (102) and form z (105) d<r = l +- 2 ds 2 + 2dsdt + dt\ where t is supposed to be the expression in s obtained from (104), and p is the radius of curvature of the curve (7, of which the sur face is the tangent surface. This result is true whatever be the relation (104). Hence equation (105) gives the element of length of any curve on the surface, and do- is called the linear element of the surface. in equations (102) has a positive or negative value, the point lies on the portion of the tangent drawn in the According as t TANGENT SUBFACE OF A CUKVE 43 in the opposite direction.
20. For the further discussion of the properties of curves it is necessary to introduce certain curves and surfaces which can be associated with them. However, in con sidering these surfaces we limit our discussion to those properties which have to do with the associated curves, and leave other con siderations to their proper places in later chapters. The totality of all the points on the tangents to a twisted curve C constitute the tangent surface of the curve. As thus defined, the sur face consists of an infinity of straight lines, which are called the P on this surface lies on one generators of the surface.
A Treatise on the Differential Geometry of Curves and Surfaces (Dover Phoenix Editions) by Luther Pfahler Eisenhart