By C. E. Weatherburn

Initially released in 1930, because the moment of a two-part set, this informative and systematically equipped textbook, essentially aimed toward college scholars, features a vectorial therapy of geometry, reasoning that by means of such vector equipment, geometry is ready to be either simplified and condensed. subject matters coated contain Flexion and Applicability of Surfaces, Levi-Civita's idea of parallel displacements on a floor and the idea of Curvilinear Congruences. Diagrams are incorporated to complement the textual content. offering a close assessment of the topic and forming a high-quality beginning for examine of multidimensional differential geometry and the tensor calculus, this publication will turn out a useful reference paintings to students of arithmetic in addition to to somebody with an curiosity within the heritage of schooling.

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Extra resources for Differential Geometry of Three Dimensions

Example text

Is the edge of The same may be proved analytically as follows. At a point r on the curve the equation of the osculating plane is ............... b = s. On differentiating with rospect -t-b-r (R -r>n = 0, (R-r)n = ....................... (12), where r and b are functions of to s we have thatis is the equation of the rectifying plane. Thus the characterbeing given by (11) and (12), is the intersection of the osculating and rectifying planes, and is therefore the tangent to the which istic, curve at r.

TWO-PARAMETER FAMILY OF SURFACES 21. Envelope. Characteristic points. An equation of the form F(a: y z a ) ) ) > b)=Q .............. (22), m which a and 6 are independent parameters, represents a doubly infinite family of surfaces, corresponding to the infinitude of values of a and the infinitude of values of 6 On any one surface both a and The curve 6 are constant. (23). These are called characteristic points, and the locus of the characteristic points is called the envelope of the family of surfaces.

Tt ; limiting position the centre of spherical curvature is the limiting position of intersection of three normal planes at adjacent points. Now normal plane at the point r is (s-r)-t = i 1 ........................ (i) s being the current point on the plane The limiting position the line of intersection of this plane and an adjacent normal pis and the equation obtained by is determined by it with respect to the arc-length (i) r)n -1 = K (a or its equivalent differentiati (cf Arts. 15, 19) s, viz.