Curve, a line described by a point moving so that the direction changes at every instant; and in mathematics the term curvature is restricted to lines that follow some law in their change of direction. Thus, the law of the circle is, that all points of it are equally distant from a fixed point, called the centre. The law of a plane curve is generally expressed by an equation between the co-ordinates of any point in it referred to a fixed point; and thus the doctrine of curves becomes matter of algebra (see CO-ORDINATES). When the equation of a curve contains only powers of and , the curve is algebraic; when the equation contains other functions, logarithms, for instance, of and , the curve is called transcendental. The cycloid, e.g., is a transcendental curve.
There are also curves, like some spirals, that do not continue in one plane; these are called curves of double curvature, and require, in analysis, three co-ordinates and two equations.—Curves are said to be of the first, second, third, &c. order, according as their equations involve the first, second, third powers of or . The circle, ellipse, parabola, and hyperbola are of the second order of curves. There is only one line of the first order—viz. the straight line, which is also reckoned among the curves.—The higher geometry investigates the amount of curvature of curves, their length, the surface they inclose, their tangents, normals, asymptotes, evolutes and involutes, &c.
The number of curves that might be drawn is of course infinite. A large number have received names, and are objects of great interest to the mathematician—in some cases, for their beauty, in others, for their geometrical properties. Among the most interesting are the following: (1) Circle; (2) ellipse; (3) hyperbola; (4) parabola; (5) cissoid of Diocles; (6) conchoid of Nicomedes; (7) lemniscata; (8) cycloid; (9) harmonic curve; (10) trochoid; (11) the witch; (12) cardioid; (13) curves of circular functions—e.g. curve of sines; (14) the logarithmic curve and other plane spirals, such as that of Archimedes, the lituus, the reciprocal or hyperbolic, and the involute (q.v.) of the circle; (15) the catenary; (16) the tractrix; (17) the tractrix; (18) the ovals of Cassini.