Surface

A general point 0 of a double curve is a binode of a special kind. The two sheets of f are quite distinct ; every simple point of f near 0 belongs definitely to one sheet or the other, the points of the double curve belonging to both. There is no continuous pass age at all from one sheet to the other near 0; but on the double curve there lie a certain number of pinch-points, unodes of a special kind (which may be absorbed in higher singularities), at which the two tangent planes coincide, and the two sheets become connected.

A tacnode of the surface is a double point where the general plane section has a tacnode, which is the plane singularity equivalent to two adjacent double points. Thus 0 has an infinite set of double points of f adjacent to it, one in every direction in the tangent plane, which may be considered as an infinitesimal double curve adjacent to 0. In a similar way, a tacnodal curve is equivalent to two adjacent ordinary double curves.

In general, a singular point, whether isolated or lying on a singular curve, is characterized, first, by its multiplicity, i.e., the number of intersections of f with a general line through the point that are absorbed there; next, by the nature of the tangent cone and its sheets, and of the singularity of a general section; and further by the nature and arrangement of the higher singularities of special sections. By a series of suitable transformations, any singular point or curve of a surface can be analysed into an equivalent set of component singular points and curves of standard types, and any surface can be transformed into one having only elementary singularities, and in particular into one having no isolated singular points, but only ordinary multiple curves and such points of higher singularity as necessarily lie on them ; or alternatively into a surface having only a double curve and a number of triple points which are also triple for the double curve. The complete theory is complicated. Isolated singular points may give rise on transformation to singular curves ; and a singular curve may give rise, from the transformation of its special points of higher singularity, to other singular curves, whose general points may be of more complicated nature (though not of higher multiplicity) than the general points of the original curve.

Singular tangent planes are defined dually to the above. Any

surface of degree > 3 must have singularities of one or other kind, or of both. A surface f of degree n with no singular points is of class n'= n(n— a surface of class ie with no singular planes is of degree n=n'(n/— These two equations are in compatible unless n=n'= 2, and f is a quadric.

If f has no point singularities, it has in general two series of double and stationary tangent planes, and its reciprocal has a double and a cuspidal curve. The characteristic numbers of these modify the equations relating degree and class, and make them compatible. If f has higher singularities, there exists a complicated set of relations between the numerical characteristics ; the simplest are the Plucker equations for the general plane section and tangent cone.

Curves on a Surface.

The interest of f depends partly on the simpler curves lying on it. There are always its plane sections, and its curves of total intersection with other surfaces; lower curves may arise as partial intersections, components of a total intersection that breaks up through acquiring additional singu larities, the cutting surface either touching f or passing through one or more of its singularities. All curves on a plane are its total intersections, with cones standing on the curves. On a quad ric, the conic section by a tangent plane acquires a double point at the point of contact and breaks up into a pair of generators. A cubic surface contains 27 pencils of conics, cut out by planes through the 27 lines, and many other systems of curves of degrees 3, 4, and 5, cut out by quadrics containing various sets of the lines.

Other important curves on f are defined by infinitesimal prop erties. An asymptotic curve is one whose tangent at any point is one of the inflexional tangents, parallel to an asymptote of the indicatrix. A curve of curvature is one whose tangent at any point is a principal tangent, parallel to an axis of the indicatrix. Two curves of each of these kinds can be drawn through any general point of f. A geodesic is a curve whose small arcs are each the shortest distance on the surface between their extremities. On a plane, these are lines ; on a developable surface, curves which become lines when the surface is flattened out. An infinite number of geodesics can be drawn through a general point of f, one starting in any direction.