If a plane through the normal meets the indicatrix in a diam eter of length 2r, the radius of curvature of the normal section is p, where = • An oblique plane meeting the tangent plane p r in the same line, and making a finite angle 4 with the normal, cuts f in a section of radius of curvature pcos4, the projection of p upon the plane (Meunier's theorem). This vanishes for the tangent plane, when the section is singular.
The major and minor axes of the indicatrix lie on the principal normal planes, making sections of maximum and minimum radii of curvature, say 132. For a normal section inclined at an angle 2 2I 0 (Euler's to the first principal section, - in — (Euler's theo P pi P2 rem).
According to the nature of the indicatrix, 0 is an elliptic hyperbolic or parabolic point of f. If it is elliptic, all the normal curvatures are of the same sign, and the surface bends away from the tangent plane on one side only. The part near 0 of the section by z is a small real closed curve for of one sign, and imaginary for of the opposite sign. The section by the tangent plane z= o has no real part in the neighbourhood except 0 itself, which is a double point with imaginary tangents. The surface is synclastic.
At a hyperbolic point, the surface is anticlastic. The principal curvatures are of opposite signs, and f bends away from the tangent plane on opposite sides in these directions. Two normal sections have inflexions, the curvatures vanishing; their planes pass through the inflexional tangents at 0, which are parallel to the asymptotes of the indicatrix, and touch the section of f by the tangent plane, which has two real branches through 0 along which f crosses the plane. All sections near 0 have real branches there. If there is a line lying on f, it is one of the inflexional tangents at each of its points. Conjugate tangent lines are those parallel to conjugate diameters of the indicatrix.
At a parabolic point, the inflexional tangents coincide; the section by the tangent plane has a cusp. These points lie on the Hessian.
The normal at 0 does not in general meet the normal at an adja cent point, unless this lies on a principal section, in which case the normals intersect at one of the two centres of curvature of f at 0. These coincide if the indicatrix is a circle; then 0 is an umbilic, all normal sections have the same curvature and all adjacent normals meet the normal at 0.
sphere (q.v.). Every plane section is a conic. Parallel sections are similar and similarly situated conics, whose centres lie on a line, a diameter of q. The diameters concur at the centre (which may lie at infinity), which bisects every chord of q through it.
There are three mutually perpendicular planes of symmetry, the principal planes, intersecting by pairs in the principal axes which meet q in its vertices.
Through each point P of q there pass two generators, real or imaginary, lying wholly on q; their plane is the tangent plane to q at P, cutting q in this pair of lines. The generators fall into two systems, any one generator meeting all those of the other system and none of its own. A quadric is determined by any three generators of one system, and may be defined as the locus of a line meeting three fixed skew lines. There are six directions (only two real) of planes of circular section; the points of contact of the tangent planes parallel to these are the umbilics.
A quadric has three focal conics (two real), to each point S of which there corresponds a directrix 1, such that the distance of any point of q from S is proportional to its distance from I meas ured parallel to one of the planes of circular section ; S and I are focus and directrix of the section of q by the plane S/, which is the normal plane at S to the focal conic. A focus can also be defined as a sphere of zero radius touching q twice, the directrix being the chord of contact.
A quadric is fixed when its three principal sections are given, i.e., it is determined by its centre, trihedron of principal planes, and lengths of principal axes. It depends on nine para meters. If the centre is finite, there are three main types of quadric, according to the types of the principal sections: ellipsoid (three ellipses), hyperboloid of one sheet (one ellipse, two hyper bolas), hyperboloid of two sheets (two hyperbolas, one section imaginary). If the centre is at infinity, one principal section is wholly at infinity; the other two are parabolas, meeting in the one finite vertex, and q is an elliptic or hyperbolic paraboloid according as these parabolas lie on the same or opposite sides of the tangent plane at the vertex. If the principal axes vanish, q is a quadric cone ; if also the centre is at infinity, it is a cylinder. Finally, a degenerate quadric is a pair of intersecting, parallel or coincident planes.