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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The equations 23 and 4 are the vector equivalents of the Serret-Frenet formulae. The ruled differential 2 of the con- gruence will be a developable if consecutive generators intersect. Orthogonal systems of curves. Rotation of n Parallel surfaces. Curvilinear coordinates on a surface. Curves on a surface; 5. The position vector r, of a point P relative to the origin 0, is the vector whose magnitude is the length OP, and whose direction is from 0 to P. If a geodesic on a surface of revolution cuts the meridians at a con- stant or, the surface is a right oyhnder The curvature of a curve, as defined in Art.

If two skew surfaoes have a common generator, and their tangent planes at three points of it are inclined at the tthree angle, they will be inclined at this angle at every point of the common generator.

It was shown in Ex. Diffsrential the torsion of a geodesio varnishes where it touches a line of curvature. This may be done as follows.

Differential Geometry of Three Dimensions – C. E. Weatherburn – Google Books

This is the equation of the tangent plane. Small deformation of a congruence of curves. Hence it contains the tangent at P to the curve PQ, and therefore also the normal at A to the first sheet of the centro-surface. Normals Curvilinear coordinates Fundamental magnitudes The mutual moment of two given generators and their shortest distance apart are clearly independent seatherburn the curve chosen as directrix. The curvature of a geodesic is the normal curvature in its direc- tion.


But QQ ‘ is weathsrburn to d’ x d and is therefore perpendicular to d’. Looking for beautiful books? If r is a pomt on the line of curvature, n is the corresponding point on the spherical image.

The Cartesian coordinates as, y, z of the point are then known functions of u, v, and the elimination of the two parameters leads to a single rela- tion between oo, y, z which is usually called the equation of the surface. For the curve a? We have seen that the lines of curvature on a surface S do not in general correspond with those on its centro-surface.

If now we consider all members of that family of lmes of curvature to which G belongs, the locus of their edges of regression is a surface, which is one sheet of the surface of centres.

But these three tangents are all m the plane PRB.

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In making the initial points and the principal planes there coincident, we altered only the position and orientation of the curves in space; and the theorem has thus been proved.

It is highly desirable that the study of the geometry of Euclidean 3-space should thus come first, and this can be undertaken with most students at an earlier stage by vector methods than by the Ricci calculus. That is to say, the lines of curvature on a paraboloid are the curves m which it is cut by the confocals. If a curve is drawn on a ruled surface so as to intersect all the generators, then, provided it has two of the following propertiesit will also have the third: If we consider the possibility of the asymptotic lines of the surface S corresponding with those of the first sheet of the evolute, we sefek to identify 10 with the equation of the asymptotic lines of the dimensuons 8.


Prove that the rectifying developable of a curve is the polar develop- able of its involutes, and conversely. Every line through P m this plane is a normal to the curve. Further, null lines on a minimal surface become both null lines and asymptotic lines m the spherical representation.

Thus h has the same sign as the principal radii of curvature, a beometry J3. These are the necessary and sufficient conditions that the parametric turves be lines of curvature.

Differential Geometry of Three Dimensions: Volume 1

Examine the sphenoal indioatnx of the prmcipal normal of a given curve The vector in brackets is tangential to the surface ; and consequent! It is usually convenient to choose as the scalar parameter the length fimensions of the arc of the curve measured from a fixed point A on it.

Isometric parameters and curves Curl of a vector Veotor functions cont. The whole second curvatwre of a geodesio triangle is equal to the excess of the sum of the angles of the triangle over two right angles.

Normal curvature Conjugate direction. Just as in equation 2 the scalar k measures the arc-rate of turning of the unit vector bby, so here t measures the arc-rate of turning of the unit vector b. The torsion is found by differentiating this.

Differential Geometry of Three Dimensions: Volume 2 : C. E. Weatherburn :

Cambridge University PressApr 15, – Mathematics – pages. Consider next the surface B which is derived from a given surface 8 by inversion. Moreover the equation is linear in each of the ratios du: