Full text of “Weatherburn C. E. Differential Geometry Of Three Dimensions Volume 1 ” Curvature of normal section MeUnier’e theorem Examples IV. Differential Geometry of Three Dimensions, Volume 2 C. E. Weatherburn of the unit vectors a b n. 7. Other differential invariants. 8. e. Differential Geometry Of Three Dimensions by. C. E Weatherburn File Type: Online Number of Pages Description This book describes the fundamentals of.
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Only occasional reference will be made to null lines. Thus the character- istic, being given by 11 and 12is the intersection of the oscu- lating and rectifying planes, and is therefore the tangent to the curve at r. If a plane cuts a surface everywhere at the same angle, the seotion is a line of ourvature on the surface.
Hence this tangent is perpendicular to d’. Directions on a surface. Diagrams are included to supplement the text. Its magnitude is the modulus of n’. Three forms of ds for pseudospherical surfaces. Weathegburn point P is then called a hyperbolic point, or a point of negative curvature.
It is denoted by — b. The remainder of the book is devoted to differ- ential dimfnsions for a surface and their applications. The quadrics confocal with this are given by a? With the notation of Art. The hne of striction of a skew surface is an orthogonal trajectory of the generators only if the latter are the bmormals of a curve, or if the surface is a right oonoid. Hence in general the spherical differsntial is not a conformal representa- weattherburn. Ratings and Reviews 1 1 star ratings 1 reviews.
Now the first weatnerburn in this equation vanishes, because, by Art 35, t is perpendicular to the rate of change of the unit normal m the conjugate direction, and an asymptotic direction is self-conjugate.
Thus at each point of the surface there are two principal curvatures, K a and Kb, and these aie the normal curvatures of the surface m the directions of the lines of curvature. Show that the lines of curvature of the helicoid in Ex.
Take the ourve thred as directrix, and let t, n, b be its unit tangent, principal normal and bmormal respectively. The parameter ahowever, determines a particular mem- ber of a family of surfaces, and has the same value at all points of that member.
The curve of intersection is thus a line of curvature on the second surfaoe also We’re featuring millions of their reader ratings on our book pages to help you diferential your new favourite book.
If two skew surfaces have a common generator and touch at three points along it, they will touch at every point of italso the central poini and the parameter of distribution of the generator are the same for both surfaces.
Differential Geometry of Three Dimensions: Volume 1
Moreover, the curves on either sheet of the evolute which corTespond to the lines of curvature on 8, form a conjugate system. Write down the value of their torsions.
Involutes of a surfaco Inverse Surfaces 82 Inverse surface. And, of all geodesics through a given point, those which bisect the angles between the lmes of curvature have the greatest torsion.
Differential Geometry of Three Dimensions – C. E. Weatherburn – Google Books
Similarly if m is constant and the curve is a line of curvature, r must vanish, and the curve is plane. Thus, if two directions are conjugate at a point on a given surface, dimensionns is perpendicular to the spherical image of the other at the corresponding point.
Prove otherwise that the inclination of conjugate lines is equal or supplementary to that of their Bphenoal image. Thus dimensiobs parameters are positive, and the values dimenssions u and v are separated by — c. Then its Cartesian coordinates as t y, z are also functions of the same parameter When the curve is not a plane curve it is said to be skew, tortuous or twisted We shall confine our attention to those portions of the curve which are free from singularities of all kmds.
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Ruled surfaces in general, however, do not possess these properties. Alternative method e e e. Further, an asymptotic line is self-conjugate. The vector in brackets is tangential to the surface ; and consequent! The unit dyadic Reciprocal dyadics.
If k is zero at all points, the curve is a straight line If r is zero geomerty all points, the curve is plane. A ourve is traced on a right circular cone so as to out all the generating lines at a constant angle. The second is the reciprocal of the length of the normal intercepted between the curve and the axis of rotation.
Prove the formulae 19 of Art. This agrees with the fact that a straight line is the path of shortest distance between two given points on the plane.