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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It may also be proved as follows. This is the required condition that 9 may represent a develop- able surface. Hence along any one geodesic the normal curvature varies as diensions cube of the central perpendicular on the tangent plane. Since the line joining any point X, Y, Z on the tangent plane to the point diffeeential contact is perpendicular to the normal, it follows that. 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 length of the arc of the geodesic joining the two points is 6 — a. The operator V for a given dimensiohs. Prove the formulae 19 of Art. Geodesio differentkal and hyperbolas From these relations several interesting general pioperties maybe deduced. Hence, in order that the asymptotic lines on S may correspond to those on each sheet of its evolute, the specific curva- ture of S must be constant.

Gekmetry, smce consecutive generators are consecutive tangents to the edge of regression, the osculating plane of this curve is that plane of the family which contains these generators But this plane touches the developable.

They are identical when K is zero In the last case the surface is a developable, and the single asymptotic line through a point is tho generator.

Thus the surfaces cut at a constant angle. And, conversely, when the linear element is of this form, the para- metric curves are systems of geodesic parallels.


Geo- desics on a sphere are great circles, and therefore plane curves. What are the first and second curvatures for the spherical image?

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The curvature of the curve at any point is the arc-rate of rotation of the tangent. They are therefore bginary on a real surface, and their importance is chiefly analytic, a differential equation of the null hnes is obtained by equating sero the square of the linear element.

The formulae 17 of Art. Surface of a sphere. Show that the equation of a paraboloid, aa? This equation represents a plane through the centre of curvature perpendicular to the principal normal It intersects the normal plane in a straight line through the centre of curvature parallel to the binomial Fig 4. For a surface given by da? Other formulae for k 0.

Thus the position vector r of a point on the curve is a function of 8, regular within the range considered. On the ellipsoid k is negative for all geodesics because both parameters are negative.

Differential Geometry of Three Dimensions: Volume 1

The formula is independent of the choice of parameters. On the surface of revolution Art. Hence a geodesic is sometimes defined as the path of shortest distance on the surface between two given points on it. Then, by the f first two results of Art.

But we have seen ee the tangents to these at corresponding points are parallel; and therefore dn. And, since d is the unit normal to the sphere, we have 3. We shall prove that, for a dmensions value of c, the locus of these involutes is a surface S cutting orthogonally all the tangents to the family of geodesics.

Parameter other than a. These may be established as follows. The Best Books of Now the first term 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.


This expression for the torsion of a geodesic is identical with that found in Art. On differentiating with rospect to s we have -t. Originally published inas the second of a two-part set, this informative and systematically organized textbook, primarily aimed at university students, contains a vectorial treatment of geometry, reasoning that by the use dimnesions such vector methods, geometry is able to be both simplified and condensed. Therefore of all sections through two consecutive points P, Q on the surface, the normal section makes the length of the arc PQ a differenttial But this is the arc of the geodesic weathsrburn P, Q.

Now, in virtue of 3 we may write this a du. They are analogous to plane polar coordinates. The first of these is equivalent to d!

The locus of the centre of geodesio curvature of a line of curvature is an evolute of the latter 2 5. On the surfaces m Ex 1 and Ex. The normal curvature is zero m the directions of the asymptotes.

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The orthogonal tra- jectories of the yhree correspond to the lines on E along which one of the principal radii of curvature is constant. Find the area of an element of a quadnc bounded by four lines of curvature If r is a pomt on the line of curvature, n is the corresponding point on the spherical image. The cen of a sphere through P and an adjacent point Q on the curve Fig lies on the plane which is the perpendicular bisector of PQ ; a the limiting position of this plane is the normal plane at P.