SURFACE (OE, Fr. surface, from Lat. superficics, upper side, surface, from super, above facics, form, figure, face). The boundary between two portions of space. As a point in a plane is determined in general by two inter secting lines, so a point in space is in general determined by three intersecting surfaces. These surfaces may be plane. quadric., or of higher order according as their equations are of the first. second, or higher degree in the linear coordinates of the system. Thus in Cartesian coordinates (see COORDINATES) the general equa tion of the first degree in x, z, or ax + by + cz + d = 0, is represented by a plane. The gen eral equation of the second degree in x, y, z, or ax' + by' + + 2fyz + 2gzx + 2hxy + 2kx + 2my +2nz + d = 0, is represented by a coni coid, or surface of the second order, also called a quadric surface. By a suitable transformation of coordinates the general equation of the second degree may be transformed into one or the other of the forms (1) As' + By= + C? = D or (2) Ai" + By' = CZ. Surfaces having the symmetric equation (1) are symmetric with respect to the origin as a centre and are called central quad rics. Non-central quadrics are included in equa tion (2). If A = B = C, equation (1) takes the form y' -F = K (= r"), the equation of the sphere (q.v.). The general equation (I) represents either an ellipsoid (q.v.) or an hyper boloid. If D = 0, and A, B, C are not all posi tive, equation (1) represents a conical surface whose vertex is at the origin. Equation (2) is represented by the surface of a paraboloid (q.v.).
A surface through every point of which a straight line may be drawn so as to lie entirely in the surface is called a ruled surface. Any one of these lines which lies on the surface is called a generating line of the surface. The cylinder, cone, hyperboloid of one sheet, colloid (q.v.), and the hyperbolic paraboloid (see PARA BOLOID) are ruled surfaces. There are two dis tinct classes of ruled surfaces, those on which the consecutive generators intersect and those on which they do not. The former are called developablo and the latter skew surfaces.
If the degree of the equation f (x, y, z) = 0 is higher than the second, the surface representing it will be of an order higher than the second. In discussing the properties of such surfaces, especially the nature of the surface in the vicin ity of any given point, the equation of the tan gent plane at that point is necessary. This plane
is the locus of all tangent lines through the given point, and will meet the surface of the nth order in a curve of the nth degree, since each straight line meets this curve in n points. The point of contact of the plane with the surface will be a singular point on the curve. (See CURVE.) The section of any surface by a plane parallel and indefinitely near to the tangent plane at any point is a conic and is called the indientrix at the point. Thus points of a surface are called ellip tic, parabolic, or hyperbolic, according as the indicatrix is an ellipse, parabola. or hyperbola. If every straight line through a point (x', y', z') of a surface meets the surface in two coincident points, the point (e, y', z') is called a singular point. If the tangent lines at any point form a cone the point is called a conical point; if they form two planes the point is called a nodal point. Similar to the envelope of a family of curves, the envelope of a family of surfaces is the locus of the ultimate intersections of a series of sur faces produced by varying one or more param eters (q.v:) of an equal ion. The curve in which any surface is met•hy the consecutive surface is called the characteristic of the envelope. Every characteristic will meet the next in one or more points, and the locus of there is called the edge of regression or cuspidal edge of the envelope. The conditions for convexity and concavity, dif ferent orders of contact, and various other prop erties are best obtained from works on analytic geometry.
Consult: Monge, Application dr l'analyse a la Peometrie (Paris, 1705) : Dupin. Developpements de (lb., 1813) Pliicker, Yrue Geome tric des Thames auf die Betrachtung der geraden Linie als Raumelement (Leipzig, 1868) ; Salmon, Analytic Geometry of Three Di mensions (4th ed., Dublin, 1882); Smith, An Ele mentary Treatise on Solid Geometry (3d ed., New York, 1S91) ; Gauss, "Allgemeine Fffichenthe orie," in Ostwald's Klassiker der esakten Wisscn schaften (Leipzig, 1900) ; Knoblanch, Li-nleitung in die allgemeine Theoric der krummen nacho'. (ib., 1888). For a brief sketch of the history of the subject, consult Smith, "History of _Modern Mathematics," in Merriman and Woodward's Higher Mathematics (New York, 1896).