Surface

The hyperboloid of one sheet and hyperbolic paraboloid have real generators, and models can be constructed by threads or wires; they are everywhere anticlastic. The ellipsoid, hyper boloid of two sheets and elliptic paraboloid have imaginary generators, and are everywhere synclastic. Referred to its prin cipal axes, an ellipsoid of semi-axes, a, b, c has the equation x2 y2 z2 = I. An elliptic paraboloid referred to its vertex as origin can have the equation 5 ± = and the other types of quadric can be represented by equally simple equations.

Cubic Surfaces.

A general cubic surface f is of class or lower if it is singular. It possesses 27 lines lying on it, of which each meets I o others, arranged in 5 intersecting pairs. There are 45 triangles lying on f, whose planes are tritangent, touching it at the vertices. We can also select, in 36 ways, a double-six from the 27 lines, consisting of two sets of six lines, each line meeting five of the other set and none of its own. Any plane through a line 1 cuts f in a residual conic meeting 1 in two points, at which the plane touches f. For two such planes, the conic touches 1, at a parabolic point of f, lying on the Hessian.

There can be as many as four double points on a cubic surface. If these lie at the vertices of the tetrahedron of reference, the equation can take the form I -I- + I = o. Each of the six x y z w edges of the tetrahedron absorbs four of the 27 lines, and there is one triangle of distinct lines, each meeting two opposite edges of the A cubic surface can have one double line; it is then skew, each plane through the double line meeting it in a residual generator. Two of these pass through each point of the double line, and their plane meets f in a fixed line of the surface, the simple directrix.

Singularities.

A singular point of a locus is one at which there is no definite, single tangent plane. If the origin 0 is singular, and as above • • • -Fun, the four conditions are those involved by uo= 0, The simplest case is the conic node, an ordinary double point represented u3+ -Fun, where is a general quadratic function of x, y, z, giving the proper tangent cone.

Any line through 0 meets f in two points coinciding at 0, a generator of u2 meets it in three points, and the six lines of closest contact given by o meet it in four. The section of f by a general plane through 0 has a double point with two quite distinct branches, the tangents being generators of u2 ; that by a plane through a line of closest contact has an inflexion on one branch.

The section by a plane touching the tangent cone has a cusp, the two branches merging into one cycle.

If a general line through 0 meets f in s points there, in the simplest case 0 is an ordinary s-fold point with a proper tan gent cone where f = . • • ; this involves i-s(s+ I) (s+ 2) conditions. Any plane through 0, or any surface having 0 as an ordinary point, meets f in a curve having an s-f old point there. If 0 is old on the second surface, the two tangent cones having no common sheet, the curve of intersection has ssi branches through 0. If 0 is also s2-f old on a third surface, the three have ssis2 points of intersection absorbed at 0.

If the tangent cone breaks up, the simplest case is the binode, a double point whose quadric tangent cone consists of two biplanes, intersecting in the edge. The section by a general plane through 0 has a double point as for a conic node, that by a plane through the edge has a cusp, and that by one of the biplanes has a triple point. Taking the biplanes as planes of reference, we have = xy; then the equation says that either x or y is small of second order, or else both are small of orders between i and 2. A general point of f near 0 is close to a definite one of the two biplanes, and, in this sense, we can speak of two sheets of f, one touched by each biplane ; they are not separate sheets, but become indis tinguishable near the edge x = y = o. There are three lines of closest contact in each biplane; these are in general different from the edge, which meets f in three points only.

The conic node and binode are ordinary singularities ; i.e., the tangent cone, proper or degenerate, has no repeated sheet. At an extraordinary singularity, the tangent cone not only breaks up, but has a repeated sheet, it, say, so that s— 21). The simplest case is the unode, a double point with a repeated tangent plane, f • • • The surface has two connected half-sheets, the general section having a cusp. There are three lines of closest contact ; the section by a plane through one of these has a tacnode, that by the tangent plane has a triple point.