Modern curves are continuous images of intervals—sometimes wild ones
Euclid called a line a breadthless length; curved lines were the flexible kin of straight ones. Today a curve is often the continuous image of an interval in a space. That definition admits monsters: space-filling and fractal curves that refuse ordinary drawing.
A curve is the trace of a moving point, the image of an interval under a continuous map into some space, and the map is called its parametrization. That broad class includes shapes that defy drawing, so mathematicians often demand a differentiable map to keep curves well behaved. Level curves and algebraic curves fall outside it and are called implicit curves because equations define them. A plane algebraic curve is where a two-variable polynomial vanishes; allow complex solutions and the result is topologically a surface, a Riemann surface, and algebraic curves over finite fields now underpin much modern cryptography.
People drew curves for decoration long before anyone studied them, as easily as trailing a stick through sand. Older English called any such path a line, so straight line and right line were needed for the unbent kind. Euclid defined a line as breadthless length, and later commentators separated determinate curves that close, such as the circle, from indeterminate ones that run forever, such as the parabola.
Greek geometers invented special curves to attack problems compass and straightedge could not solve. Diocles used his cissoid to double the cube, Nicomedes his conchoid to double the cube and trisect angles, and Archimedes his spiral to trisect angles and square the circle, while Perseus studied slices of tori. René Descartes changed the field in the seventeenth century by describing curves with equations, which split algebraic curves from transcendental ones.
Calculus opened more doors. Kepler applied conics to astronomy, the brachistochrone and tautochrone problems led to the cycloid, and the catenary is named for the shape of a hanging chain. Newton sorted cubic curves into ovals. Since the nineteenth century curves have been treated as the one-dimensional case of manifolds, yet puzzles such as the Jordan curve theorem and Hilbert's sixteenth problem remain their own.
Source: Curve
