Overview

Some curve-defining techniques, such as those using the Hermite basis functions, interpolate a given set of points. This means that the curve produced passes exactly through the points. Other techniques define a curve that only passes near or approximates the given points. Interpolation techniques like the Hermite basis have certain disadvantages when incorporated into an interactive geometric modeling system. They usually do not give the user an intuitive sense of how to change or control the shape of a curve. For example, changing the shape of a spline-interpolated curve by moving one or more of the interpolating points may produce unexpected, if not undesirable, perturbations and inflections, both locally and globally. To control a curve s shape in a predictable way by changing only a few parameters is more desirable. The B zier curve partially satisfies this need.

P.B zier, who was familiar with the work of Ferguson and Coons and their parametric cubic curve and bicubic surface interpolating techniques, in the early 1960s set out to find what he hoped would be a mathematical format more amenable to the designer and the design process. The result of his work was the UNISURF system, used by the Renault Company in the 1970s to design the sculptured surfaces of many of its automobile bodies. At the heart of the UNISURF system were the curves and surfaces that bear his name.

B zier started with the principle that any point on a curve segment must be given by a parametric function of the following form: