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SURFACE
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SURFACE , the bounding or limiting parts of a See also:body. In the See also:article See also:CURVE the mathematical question is treated from an See also:historical point of view, for the purpose of showing how the leading ideas of the theory were successively arrived at. These leading ideas apply to surfaces, but the ideas See also:peculiar to surfaces are scarcely of the like fundamental nature, being rather developments of the former set in their application to a more advanced portion of See also:geometry; there is consequently less occasion for the historical mode of treatment. Curves in space are considered in the same article, and they will not be discussed here; but it is proper to refer to them in connexion with the other notions of solid geometry. In See also:plane geometry the elementary figures are the point and the See also:line; and we then have the curve, which may be regarded as a singly See also:infinite See also:system of points, and also as a singly infinite system of lines. In solid geometry the elementary figures are the point, the line and the plane; we have, moreover, first, that which under one aspect is the curve and under another aspect the developable (or torse), and which may be regarded as a singly infinite system of points, of lines or of planes; and secondly, the surface, which may be regarded as a doubly infinite system of points or of planes, and also as a See also:special triply infinite system of lines. (The tangent lines of a surface are a special complex.) As distinct particular cases of the first figure we have the plane curve and the See also:cone, and as a particular See also:case of the second figure the ruled surface, See also:regulus or singly infinite system of lines; we have, besides, the congruence or doubly infinite system of lines and the complex or triply infinite system of lines. And thus crowds of theories arise which have hardly any analogues in plane geometry; the relation of a curve to the various surfaces which can be See also:drawn through it, and that of a surface to the various curves which can be drawn upon it, are different in See also:kind from those which in plane geometry most nearly correspond to them—the relation of a system of points to the different curves through them and that of a curve to the systems of points upon it. In particular, there is nothing in plane geometry to correspond to the theory of the curves of curvature of a surface. Again, to the single theorem of plane geometry, that a line is the shortest distance between two points, there correspond in solid geometry two extensive and difficult theories—that of the geodesic lines on a surface and that of the minimal surface, or surface of minimum See also:area, for a given boundary. And it would be easy to say more in See also:illustration of the See also:great extent and complexity of the subject. In See also:Part I. the subject will be treated by the See also:ordinary methods of See also:analytical geometry; Part II. will consider the Gaussian treatment by differentials, or the E, F, G See also:analysis. Additional information and CommentsThere are no comments yet for this article.
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