An elementary treatise on coordinate geometry of three dimensions . n (1) is satisfied; there-fore the normals at adjacent points of a line of curvatureintersect. If the normals at O and P intersect, cos 0 = 0 or sin 0 = 0,and therefore O and P are adjacent points of one of theprincipal sections, or the curve is a line of curvature. Ex. The normals to an ellipsoid at its points of intersection witha confocal generate a developable surface. §§229-231] LINES OF CURV VTURE :;.:• 231. Lines of curvature on a surface of normals to a surface ol revolution at all points of ameridian se
An elementary treatise on coordinate geometry of three dimensions . n (1) is satisfied; there-fore the normals at adjacent points of a line of curvatureintersect. If the normals at O and P intersect, cos 0 = 0 or sin 0 = 0,and therefore O and P are adjacent points of one of theprincipal sections, or the curve is a line of curvature. Ex. The normals to an ellipsoid at its points of intersection witha confocal generate a developable surface. §§229-231] LINES OF CURV VTURE :;.:• 231. Lines of curvature on a surface of normals to a surface ol revolution at all points of ameridian section \\>- in the plane of theby §230, the meridian sections are lines of curvature Thenormals at all points of a circular section pass through thesame point on the axis, and therefore any circular section isa line of curvature. Let P, (tig. G3), be any point on the surface, and let PTand PK be the tangents to the meridian and circular sectionsthrough P. Let PN be the normal at P, meeting the axisin N, and let C be the centre of the circular section. Then. Fig. (53. TPN and KPN are the planes of the principal sections. Theprincipal radius in the plane TPN is the radius of curvatureat P of the generating curve. The circular section is anoblique section through the tangent PK. and its radius ofcurvature is CP. Therefore, by Meuniers theorem, if p isthe principal radius in the plane KPN. CP = pcos,6, where 0= p = PN. Thus the other principal radius is the intercept on thenormal between P and the axis. Ex. 1. In the surface formed by the revolution of a parabolaabout its directrix one principal radius at any point is twice the other. 336 COORDINATE GEOMETRY [ch. xvi. Ex. 2. For the surface formed by the revolution of a catenaryabout its directrix, (the catenoid), the principal radii at any point areequal and of opposite sign. (A surface whose principal radii at each point are equal and ofopposite sign is called a minimal surface.) Ex. 3. In the conicoid formed by th
Size: 1585px × 1577px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
