A Bezier curve is a parametric curve whose shape is determined by a small set of control points. The endpoints lie on the curve, while intermediate control points pull the curve toward them without being touched, giving designers an intuitive handle on smooth free-form shapes. The mathematics rests on Bernstein polynomials, and the curve can be evaluated stably by repeated linear interpolation, the construction now known as de Casteljau’s algorithm.
The technique was developed independently by two French automotive engineers in the 1960s. Pierre Bezier, at Renault, built the UNISURF computer-aided design system to describe car body surfaces; Paul de Casteljau, at Citroen, arrived at the same family of curves slightly earlier but published later because of his employer’s secrecy. The curves carry Bezier’s name because Renault allowed him to publish. One of his primary accounts is “Example of an existing system in the motor industry: the Unisurf system,” in the Proceedings of the Royal Society of London, Series A, volume 321, pages 207 to 218, in 1971 (DOI 10.1098/rspa.1971.0027), which describes how UNISURF let stylists define and manipulate surfaces numerically.
The appeal of Bezier curves is that they are compact, predictable, and easy to compute, transform, and subdivide. A few control points encode a complex smooth shape, affine transformations of the control points transform the whole curve, and de Casteljau subdivision splits a curve into two Bezier pieces for refinement or rendering. Joining low-degree Bezier segments end to end, typically cubics, produces the piecewise curves used throughout graphics.
That practicality made Bezier curves ubiquitous far beyond car design. Cubic Beziers define the outlines of PostScript and OpenType fonts, the paths in PDF and SVG and illustration programs, and the easing and motion paths in animation tools. Surface generalizations, Bezier patches, underpin much of computer-aided design and 3D modeling. A method born from shaping sheet metal became one of the most widely used representations of smooth geometry in computing.