Nonlinear Design. Bezier curves

hroughout centuries, a ruler and a pair of compasses remained the only practical devices for approximating abstract geometric figures in real-world drawings and designs. Consequently, straight lines and circular arcs were the only geometric objects that could be reliably and accurately reproduced on paper. More complex lines either required arduous calculations (in mathematical drawings), or were just drawn by hand (in visual arts and design).

In the 20th century, however, this deficiency proved to be a real obstacle, especially in industrial design where complex curvilinear lines and surfaces must be defined precisely so as to be uniformly reproduced in metal or plastic. When you think of curvilinearity in technics, it's likely that one of the first things that comes to mind are the smooth contours of modern cars. Indeed it was the French Rénault car manufacturing corporation for which Pierre Bézier developed, in 1960s, a new design tool based on one very versatile variety of mathematical curves.

These curves, named Bezier curves after their inventor, are now familiar to any user of a vector drawing program. The most commonly used Bezier curves of third order are fully defined by four points: two endpoints and two control points that do not lie on the curve itself but define its shape (Fig. 1).