## A Treatise on the Differential Geometry of Curves and Surfaces |

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A Treatise on the Differential Geometry of Curves and Surfaces Luther Pfahler Eisenhart Limited preview - 2013 |

A Treatise on the Differential Geometry of Curves and Surfaces Luther Pfahler Eisenhart Limited preview - 2004 |

### Common terms and phrases

applicable surfaces asymptotic lines axes axis Bianchi binormal circles congruence conjugate system consequently const constant angle coordinates corresponding points cyclic system cylinder Darboux defined deformation denotes the angle determined developable surface differential parameters differentiated with respect direction-cosines dudv dv dv edge of regression envelope equa equal expressions follows form a conjugate found by quadratures geodesic curvature helicoid Hence infinity integral latter line of striction linear element lines of curvature locus meridians minimal surface moving trihedral necessary and sufficient obtain orthogonal trajectories osculating plane paraboloid parallel parametric curves parametric lines perpendicular plane curve principal normal principal radii pseudospherical surface quadrics radius reducible referred Ribaucour ruled surface satisfy the condition Show solution sphere spherical representation straight line substituted sufficient condition surface applicable surface of revolution tangent plane theorem tion torsion total curvature transformation values Weingarten z-axis zero

### Popular passages

Page 219 - ... because the distances which we measure are too small, is often discussed. Gauss, in order to decide this question experimentally, measured as exactly as possible, and taking into account all possible sources of error, as large a triangle as possible (Brocken, Inselberg, Hoher Hagen), but could not find a difference between the sum of the angles of the triangle, and two right angles, which exceeded the limits of error. We shall return almost immediately to the question as to how far experience...

Page 241 - A surface which can be generated by the motion of a straight line is called a ruled surface. The infinitude of straight lines which thus lie on the surface are called its

Page 112 - In any triangle, the sum of the three angles is equal to two right angles, or 180°.

Page 58 - Find the volume of the tetrahedron formed by the coordinate planes and the plane S + 5 + S-1' (7) F'S- 127- where a, b, c are all positive.

Page 403 - if a surface of reference of a normal congruence be deformed in such a way that the directions of the lines of the congruence with respect to the surface be unaltered, the congruence continues to be normal'.

Page 121 - D'dv) = 0 , (16) R 1 gndu+gndv Dndu + D'dv (17) D'du + D&dv = 0 The directions, for which R is extremura, are obtained from the equation (17). Then the two systems of curves passing through a point on the surface determine the directions at the point for which the radii of r-normal curvature have their maximum and minimum values. These curves are called the r-lines of curvature of the first kind and their tangents at a point the r '-principal directions of the first kind at the point.

Page 209 - ... be stated as follows : The excess over 180° of the sum of the angles of a triangle formed by shortest lines on a concavo-concave...

Page 332 - On the integration of a partial differential equation of the second order of the hyperbolic type, with more than two independent variables, by MR d'Adhemar.

Page 101 - If X, Y, Z are the coordinates of a point on the hodograph, we have for the relation between the two curves, 10,NY dx „ dy

Page 56 - AC2, and B3C3 all lie in one plane which is called the tangent plane to the surface at the point A. The point A is the point of contact. Since ADt and AD2 are any curves of the surface, it follows that, in general, the tangent at A to every curve of the surface through this point will lie in the tangent plane. Therefore, the tangent plane will contain all straight lines tangent to lines Fio.