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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We may notice in passing that null lines on a surface correspond to null lines in the conformal representation.
The parametric curves clearly constitute an isometric system Art. Thus the surfaces cut at a constant angle. Family of curves continued Then r is a function of u only, but the other quantities are functions of u and another parameter v.
Goodreads is the world’s largest site for readers with over 50 million reviews. In any representation of a surface S on another, S’, the oross-ratio of four tangents at a point of S is equal to the cross-ratio of the corresponding tangents to S’ 9.
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The equation 8however, fails to determine these directions when the coefficients vanish identically, that is to say when E: It follows from the above that the direction of a line of curvature at any point is a principal direction at that point.
The locus of the point of intersection of consecutive rays on the surface is the edge of regression of the developable. 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. The locus of the ultimate intersec- tions of consecutive characteristics of a one-parameter family of surfaces is called the edge of regression.
If a vector d is localised m a line through the point whose position vector is r relative to 0the moment of d about 0 is the vector r x d. Line of stnotion Central point Modulus of extension Change of inclination e. Similarly the normal at B to the second sheet of the centro-surface is parallel to the tangent at P to the other line of curvature PR. Geodesic distanoe Hence the surface is left-handed or right-handed according as the torsion of the ourve is positive or negative.
We shall now prove that the envelope touches ENVELOPES 15] 41 each member of the family of sw faces at all points of its character- istic The characteristic corresponding to the parameter value a lies both on the surface with the same parameter value and on the envelope Thus all points of the characteristic are common to the surface and the envelope.
On the surface of revolution Art. Eut each of these terms vanishes because n’ and S’ are both parallel to t Thus oos 6 is constant, and the surfaces cut at a constant angle. If we give it that from P toward the concave side of the curve it follows from 2 that k must be regarded as positive, for t’ has also this direction.
Then the principal normal to the curve is not parallel to n.
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Conversely, if two surfaces cut at a constant angleand the foirve of intersection is a line of curvdture on one of them, it is a line of curvature on the other also. Let u be the perpendioular distance of a pomt from the axis, and v the inclination of the meridian plane through the point to the za?
They are real and different when M 1 — LN is positive, that is to say when the specific curvature is negative They are imaginary geometty K is positive. Conversely we may prove that a curve whose curvature and torsion are in a constant ratio is a helix.
Differential Geometry of Three Dimensions: Volume 2
Show that the equation of a paraboloid, aa? And since k is constant it then follows that k 0 is constant, so that the cylmder on which the helix is drawn is a circular cylinder. The parameter ahowever, determines a particular mem- ber of a family of surfaces, and has the same value at all points of that member. They are clearly equivalent to the equations 4. The normal plane at P is the plane through P perpendicular to the tangent.
Family of Weingarten surfaces.
Differential Geometry of Three Dimensions: Volume 2 : C. E. Weatherburn :
Any two Btereographic projections of a sphere are inverses of each other, the origin of inversion in either being the origin of projection m the other 8. The right helicoid is the only real ruled minimal surface. Diagrams are included to supplement the text.
Triply orthogonal thrse of surfaces; Edge of regression 42