CURVE (OF. courbc. corbe, Fr. courbe, Sp., Port., It. cum), from Lat.curvus,curved,OChureh Slay. krivii, bent, Lith. krcivas, crooked). in common language, a line that constantly de parts from a fixed direction. In analytic geome try, however, the word curve is commonly used to designate the locus of a point moving accord ing to any definite law, and hence to include the straight line. If the statement of the law according to which the point moves can be translated into an equation or equations be tween the coordinates (q.v.) of the moving point, these equations may be used to represent the curve—e.g. the circle is the locus of a point moving in a plane at a constant finite distance from a fixed point in that plane. and its equa tion is = (See COORDINATES.) If the curve possesses the property of continuity (q.v.) it is precisely definable at every point, although it may contain singularities. The form of a curve corresponds to the nature of its equation; hence a curve may be designated as algebraic or transcendental according as its equation con sists of algebraic or transcendental functions of the coordinates; for example, the collie sections are algebraic curves, and the cycloid, the loga rithmic spiral, and the catenary are transcenden tal curves. Algebraic curves are fundamentally grouped into orders and classes, according to Newton's classification. The order of a plane curve is determined by the number of points, real or imaginary, in which it intersects any line in its plane. Curves which cut such lines in two points are called curves of the second order; those which cut the lines in three points curves of the third order, and so on—e.g. the conic sections are all curves of the second order, and cubic curves are of the third order. The
straight line is the only line of the first order. Similarly the order of an algebraic curve in space depends upon the inunbor of points in which it cuts any plane. The class of an alge braic plane curve is determined by the number of tangents, real or imaginary. which can be drawn to it from any point in its plane. If two tangents are possible it is a curve of the second class, if three are possible. a curve of the third class, and so on—e.g. the collie sections are curves of the second class; the cissoid (q.v.) is of the third class. Similarly the class of a space curve is given by the number of tangent Diane; which can be drawn containing :my fixed line. The class of a plame curve depends directly upon its order when no singularities exist. If n is the order and c the class, c = n(// — 1 ). Thus a conic with Ito singular points is of second class, since c ='2(3 — 1 ) = 2; the cubic is of the sixth class, since = 3(3— 1 ) = 6. But singularities tend to diminish the class. Pliieker gave six equations connecting the order, class, number of double points, number of double t mgents, number of stationary points, and num ber of stationary tangents front whiell, if any three of these numbers are given, the other three may lie obtained. The one directly connecting the order and class is c = — a — 2d —3p, in which c is the class, it the order, d the number of double points, and p the number of stationary points. Thus. a cubic with one double point is a curve of the fourth class, since c = 9 — 3 — 2 = 4. By the aid of eovariants (see Foams), the class of a curve can be determined directly.