CURVE means, in common language, a crooked line that departs gradually from the straight direction; in mathematics, however, it is usually 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 center. 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. See CO-ORDINATES. When the equation of a curve contains only powers of x and y, the curve is algebraic; when the equation contains other functions, logarithms, for instance, of x and y, the curve is called transcendental. The cycloid, e.g., is a transcendental curve.
There are also curves, like the spiral, that do not continue in one plane; these are called curves of double curvature. To express the law of such a curve requires three co-ordinates and two equations.—Curves are said to be of the first, second, third, etc., order, according as their equations involve the first, second, third powers of x or y.
The circle ellipse, parabola, and hyperbola are of the second order of curves. There is only one line of the first order, namely, the straight line, which is also reckoned among the curves.—The higher geometry investigates the amount of C. of curves, their length, the surface they inclose, etc.
The number of calves 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 remarkable properties. Among the most interesting are the following: 1, circle; 2, ellipse; 3, hyperbola; 4, parabola; 5, cissoid of Diodes; 6, conchoid of Nicomedes; 7, lcmniscata; 8, cycloid; 9, harmonic curve; 10, trochoid; 11, the witch; 12, cardiode; 13, curves of circular functions—e.g., curve of sines; 14, the logarithmic curve; 15, the spiral of Archimedes; 16, the catenary; 17, the tractory; 18, the tractrix; 19, the ovals of Cassini; 20, the reciprocal spiral.