As a point Q moves in an arbitrary direction on a surface from coincidence with a chosen point P, the normal at it, as a rule, at once fails to meet the normal at P; but, if it takes the direction of a line of curvature through P, this is instantaneously not the case. We have thus on the normal two centres of curvature, and the distances of these from the point on the surface are the two principal radii of curvature of the surface at that point; these are also the radii of curvature of the sections of the surface by planes through the normal and the two principal tangents respectively; or say they are the radii of curvature of the normal sections through the two principal tangents respectively. Take at the point the axis of z in the direction of the normal, and those of x and y in the directions of the principal tangents respectively, then, if the radii of curvature be a, b (the signs being such that the coordinates of the two centres of curvature are z = a and z = b respectively), the surface has in the neighbourhood of the point the form of the paraboloid
| z = | x² | + | y² | , |
| 2a | 2b |
and the chief-tangents are determined by the equation 0 = x²/2a + y²/2b. The two centres of curvature may be on the same side of the point or on opposite sides; in the former case a and b have the same sign, the paraboloid is elliptic, and the chief-tangents are imaginary; in the latter case a and b have opposite signs, the paraboloid is hyperbolic, and the chief-tangents are real.
The normal sections of the surface and the paraboloid by the same plane have the same radius of curvature; and it thence readily follows that the radius of curvature of a normal section of the surface by a plane inclined at an angle θ to that of zx is given by the equation
| 1 | = | cos² θ | + | sin² θ | . |
| ρ | a | b |
The section in question is that by a plane through the normal and a line in the tangent plane inclined at an angle θ to the principal tangent along the axis of x. To complete the theory, consider the section by a plane having the same trace upon the tangent plane, but inclined to the normal at an angle φ; then it is shown without difficulty (Meunier’s theorem) that the radius of curvature of this inclined section of the surface is = ρ cos φ.
Authorities.—The above article is largely based on that by Arthur Cayley in the 9th edition of this work. Of early and important recent publications on analytical geometry, special mention is to be made of R. Descartes, Géométrie (Leyden, 1637); John Wallis, Tractatus de sectionibus conicis nova methodo expositis (1655, Opera mathematica, i., Oxford, 1695); de l’Hospital, Traité analytique des sections coniques (Paris, 1720); Leonhard Euler, Introductio in analysin infinitorum, ii. (Lausanne, 1748); Gaspard Monge, “Application d’algèbre à la géométrie” (Journ. École Polytech., 1801); Julius Plücker, Analytisch-geometrische Entwickelungen, 3 Bde. (Essen, 1828-1831); System der analytischen Geometrie (Berlin, 1835); G. Salmon, A Treatise on Conic Sections (Dublin, 1848; 6th ed., London, 1879); Ch. Briot and J. Bouquet, Leçons de géométrie analytique (Paris, 1851; 16th ed., 1897); M. Chasles, Traité de géométrie supérieure (Paris, 1852); Wilhelm Fiedler, Analytische Geometrie der Kegelschnitte nach G. Salmon frei bearbeitet (Leipzig, 5te Aufl., 1887-1888); N.M. Ferrers, An Elementary Treatise on Trilinear Coordinates (London, 1861); Otto Hesse, Vorlesungen aus der analytischen Geometrie (Leipzig, 1865, 1881); W.A. Whitworth, Trilinear Coordinates and other Methods of Modern Analytical Geometry (Cambridge, 1866); J. Booth, A Treatise on Some New Geometrical Methods (London, i., 1873; ii., 1877); A. Clebsch-F. Lindemann, Vorlesungen über Geometrie, Bd. i. (Leipzig, 1876, 2te Aufl., 1891); R. Baltser, Analytische Geometrie (Leipzig, 1882); Charlotte A. Scott, Modern Methods of Analytical Geometry (London, 1894); G. Salmon, A Treatise on the Analytical Geometry of three Dimensions (Dublin, 1862; 4th ed., 1882); Salmon-Fiedler, Analytische Geometrie des Raumes (Leipzig, 1863; 4te Aufl., 1898); P. Frost, Solid Geometry (London, 3rd ed., 1886; 1st ed., Frost and J. Wolstenholme). See also E. Pascal, Repertorio di matematiche superiori, II. Geometria (Milan, 1900), and articles now appearing in the Encyklopädie der mathematischen Wissenschaften, Bd. iii. 1, 2.
(E. B. El.)
V. Line Geometry
Line geometry is the name applied to those geometrical investigations in which the straight line replaces the point as element. Just as ordinary geometry deals primarily with points and systems of points, this theory deals in the first instance with straight lines and systems of straight lines. In two dimensions there is no necessity for a special line geometry, inasmuch as the straight line and the point are interchangeable by the principle of duality; but in three dimensions the straight line is its own reciprocal, and for the better discussion of systems of lines we require some new apparatus, e.g., a system of coordinates applicable to straight lines rather than to points. The essential features of the subject are most easily elucidated by analytical methods: we shall therefore begin with the notion of line coordinates, and in order to emphasize the merits of the system of coordinates ultimately adopted, we first notice a system without these advantages, but often useful in special investigations.