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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Thus, if idle surface is not a minimal surfacethe lines of curvature are weayherburn only orthogonal system whose spherical image is orthogonal. If the principal normals of a curve intersect a fixed line, the curve is a geodesio on a surface of revolution, and the fixed line is the axis of the surface. Thus, if two directions are conjugate at a point on a given surface, each is perpendicular to the spherical image of the other at the corresponding point.
When a one-to-one corre- spondence exiHts between the points of two surfaces, either surface may be said to be represented on the other. It is easy to show that the Vines of curvature on a surface are orthogonal in their spherical representor tion. The positive sense for a rotation about a vector is that which bears to the direction wwatherburn the vector the same relation that the sense of the rotation of a right-handed sorew bears to the direction of its translation.
Unit normals to the parametric surfaces. If two curves, 0 and G 0i touch each other at P, we may define their relative curvature at this pomt as the difference of their vector curvatures.
Related invariants of a vector function. Show that conjugate hnes are not generally inverted into conjugate lines, nor asymptotic hnes into thrree hnes. Let dashes denote differentiation with respect to the differenntial 8 of the curve. But this angle is constant.
Differential Geometry of Three Dimensions: Volume 1 : C. E. Weatherburn :
Selected pages Title Page. The point of intersection of consecutive normals along a line of curvature at P is called a centre of curvature of the surface, and its distance diffeeential P, measured in the direction of the unit normal n, is called a principal radius of curvature of the surface. On any other skew surfaoe there are in general four lines of curvature which have this property 2 1.
The surface in the fhbourhood of the point then lies entirely on one side of the jent plane at the point. If the value of the constant is altered, so in general is the surface. Thus the surfaces cut at a constant angle. Central quadrics thus belong to the class of surfaces called Liouville surfaces Art. This leads to EM – FL dv? Thus the principal curvatures at any point of the surface are equal in mag- nitude and opposite m sign, and the mdicatrix is a rectangular hyperbola.
The first term vanishes because n is perpendicular to b, and the equation then shows that differentoal is perpendicular to t But it is also perpendicular to b, and must therefore be parallel to n We may then write db ds. Show that the lines of ourvature of a surface of revolution remain isometrio in diffeential spherical representation. Prove that each sheet of the evolute of a pseudo-sphere is applicable to a oatenoid. And, of all geodesics through a given point, those which bisect the angles between the lmes of curvature have the greatest torsion.
It will be noticed that the expression [n, r’, r “] for the geodesic curvature is the same as that found in Art. In all cases one of the three surfaces through m, y t z is an ellipsoid, one an hyperboloid of one sheet, and one an hyperboloid of two sheets. It would be out of place here to attempt a systematic discussion of conformal representation.
Straight hues on a surface are the only asymptotio lines which are geodesics. Those which are not developable are called skew surfaces or scrolls.
Full text of “Weatherburn C. E. Differential Geometry Of Three Dimensions Volume 1 “
If the asymptotio lines are taken as parametrio curves, show that the ainardi-Codazzi relations become M x. This circle clearly lies in the osculatmg plane at P, and its curvature is the same as that of the curve at P, for it has two consecutive tangents in common with the curve.
The geodesics on a circular cylinder are helices. It is clear from Fig. Thus the order of the factors in a cross product cannot be changed without altering the sign of the product. Second and third of a dyadic. Providing a detailed overview of the subject and forming a solid foundation for study of multidimensional differential geometry and the tensor calculus, this book will prove an invaluable reference work to scholars of mathematics as well as to anyone with an interest in the history of education.
Find the envelope of the family of planes 3a? These are called charactenstic points, and the locus of the charac- teristic pomts is called the envelope of the family of surfaces. 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.
Differential Geometry of Three Dimensions: Volume 1
Then, by the f first two results of Art. Geoketry PA, PB are equal in magnitude to the principal radii of curvature at P The normal PA is a tangent to the edge of regression corresponding to the curve PQ, and is there- fore a tangent to the first sheet of the centro-surface at A.
diffwrential Show that the sum of the squares of the intercepts on the co- ordinate axes made by the tangent plane to the eurfaoe a;? Surface of principal normals Consider the skew surface generated by the principal normals to a twisted ourve.
It follows from the above that the direction of a line of curvature at any point is a principal direction at that point.