Surface

SURFACE. There is no agreed definition of a surface. We may think of it as the boundary of a solid body, or as the division between two portions of space, or as the locus of an set of points, selected by some law from the co' points of space, and having a certain amount of continuity; in each case a plane is the simplest example; but none of these gives a precise or satis factory definition. A very simple model illustrates the kind of difficulty that arises. If a surface partitions space, it must have two sides : but a long strip of paper, with the ends joined after one has received a half-twist, forms a surface with only one side. At an ordinary point 0 of a surface, there exists a tangent plane, the unique limit of the plane OPQ, when the adjacent points P,Q---->0 by any paths lying on the surface, the angle POQ remaining finite. Any point where this limit does not exist is singular. The normal is the perpendicular to the tangent plane at its point of contact 0.

A surface can also be regarded as the locus of a point satisfying a geometrical condition expressed by a single equation between its rectangular cartesian coordinates, say f(xyz)=o. From this point of view, the definition of a surface depends on what types of function are admissible for f, and the question of continuity arises in another form. The surface is analytic if f is an analytic function, and in particular is algebraic if f is a polynomial. We shall use f as the name of the surface whose equation is f=o. We may also consider a surface as the locus of a curve varying according to some definite law. Thus a sphere is the locus of a meridian circle which rotates about the polar axis, or of a circle of latitude whose centre advances along the axis, while the radius first increases from zero and then decreases. Any surface can be thus regarded, for it is always the locus of its own sections by a fixed pencil of planes.

In particular, a

ruled surface is the locus of a variable line, whose different positions are the generators. There are two chief kinds of ruled surface: (I) If adjacent generators do not inter sect, we have a skew surface; the simplest example is a hyperbo loid of revolution, locus of a line which rotates about an axis which it does not meet. The shortest distance between adjacent generators determines a point on each, whose locus is the curve of striction of the skew surface. (2) If adjacent generators inter

sect, the surface is developable, and the locus of the point of in tersection is the edge of regression, which is a cuspidal curve on the surface. The generators are the tangent lines to this curve. By the principle of duality, any surface is also the envelope of a continuous co' set of tangent planes, selected from the oo3 planes of space by some definite law. From this aspect, a point is the simplest example. Exceptionally, a developable surface has only co 1 tangent planes, each touching it along a generator, in stead of at a point; viz., the intersection of the plane with its con secutive. If each tangent plane is rotated about its generator of contact through an infinitesimal angle till it coincides with the consecutive plane, the whole surface is flattened out or developed into one plane, without tearing or stretching. Part of the plane is covered twice by the deformed surface, and part is uncovered, the edge of regression furnishing the boundary.

Whether regarded as a locus or an envelope, a surface

f has 003 tangent lines, defined as the limit either of the join of two ad jacent points or of the intersection of two adjacent tangent planes. To touch f imposes one condition on the lines of space, and f has in general bitangents, each touching it at two distinct points, and oo 2 inflexional tangents, each meeting f in three adjacent points.

Degree, Class, Rank. The

degree of an algebraic surface is that of its equation, i.e., that of the polynomial f. It is the degree of a plane section, and is also the number of intersections, real, coincident or imaginary, of f with any line. These are particular cases of more general properties of the degree. Two surfaces of degrees n, intersect in a curve of degree nni, or in an ag gregate of curves of this equivalent total degree, when curves of contact and common multiple curves are counted a proper num ber of times. Three surfaces of degrees n, n1, n2 have n,ni,n2 distinct points of intersection, or the equivalent of this when points of contact and common multiple points are properly counted.