SURFACE, SURFACES, THEORY OF. For the mere definition of surface, see SOLID, &c. We are here to speak of that branch of algebraic, geometry which considers the generation and properties of curve surfaces following an assigned law.
If three planes, each at right angles to the other two, be taken as the planes of CO-OR DINATES, the position of any point is determined so soon as its co-ordinates, or distances from the three planes, are given in sign and magnitude. If the co-ordinates of a point be x, y, z, and if between these one equation exists, p (x, y, 0=0, any point may be chosen in the plane of x and y, by means of given values of x and y, and the corresponding value or values of z may be found from the equation. The locus of all the points whose position can be ascertained by determining one of the co-ordinates from this equation, the other two being taken at pleasure, is a surface of which ip (x, y, 0=0 is called the equation, and the modes of proceeding are pointed out in all works on algebraic geometry. The applications of the differ ential calculus depend on the principles explained in TANGENT : the graphical use of the whole method depends mostly on descriptive geo metry, whether formally known under that name or not.
Surfaces are distinguished algebraically by the nature and order of their equations. Thus we have surfaces of the first order, in which the equation is of the first degree (this class contains the plane only); surfaces of the second order, which will be classified in the next article ; and so on.
Surfaces are also distinguished by their mode of generation, and some of the principal cases are as follows I. Cylindrical surfaces are generated by a straight line infinitely produced in both directions, which moves so as always to be parallel to a given line, and to have one of its points on a given curve.
2. Conical surfaces are generated by a straight line iofinitely pro duced in both directions, which always passes through a given point or vertex, and has one point in a given curve. The common CYLINDER and cons would be described in this science as a right circular cylinder and a right circular cone. The cylindrical surfaces themselves are only an extreme case of the conical ones, being what the latter become when the vertex is removed to an infinite distance.
2. Surfaces of rerolution are generated by the rotation of a curve about an axis, relatively to which it always retains one position. The common cone and cylinder, the SPHERE, and others of the greatest practical use, are contained in this class.
4. Tubular surfaces are generated by a circle of given radius, which moves with Its centre on a given curve, and its plane at right angles to the tangent of that curve. When the given curve is a circle, the tubular surface is a common ring.
5. Ruled surfaces (the surfaces riglees of the French writers) are those which are described by the motion of a straight line, which neither remains parallel to a given line nor always passes through a given point. This includes, among many others, the whole class of conoidal surfaces, made by a straight line which moves parallel to a given plane, and always passes through a straight line perpendicular to that plane, and also through a given curve. The surface of a spiral staircase, as it would be if there were no steps but only a gradual ascent, is an instance.
6. DerdopaUe surfaces are those which can be unwrapped on a plane without any doubling of parts over one another, or separation ; that Is, without being rumpled or torn. The only familiar instances are the cylinder and cone.