Interactive interpolation and approximation by Bezier polynomials

In computational geometry (the computer representation, analysis and synthesis of shape information*) interpolation and approximation techniques are often used for both curves and surfaces. However, the properties of 'shape' are different from the properties of functions and the well-known techniques of functional interpolation and approximation are not necessarily suitable. Shape is, for example, an axis independent phenomenon and vector valued parametric curves and surfaces are often adopted for this reason alone. Indeed, parametric notation will be used throughout this paper, both for convenience and to emphasise that the paper is concerned with the approximation of shapes rather than the approximation of functions. Sometimes, but by no means always, the functional form of a particular shape is known—a curve may be a circular arc. In such cases conventional fitting techniques may be used, but in general what is required is an acceptably close fit (to within a given tolerance) which maintains the character of the curve or surface and is smooth or fair. This paper considers the interactive design of curves, ab initio, and the interactive approximation of curves. Just as mathematical techniques are modified or developed to cope with shape, so they must be amended when interaction between a computer and a human operator is involved. A particularly elegant technique has been developed by Bezier (1968a, 1968b, 1970) of Regie Renault. This paper develops the mathematical properties of Bezier's methods for interactive approximation. In the car industry the problem is to find a mathematical representation for a stylist's clay model or sketch. The data has two basic kinds of error: measurement error and error due to the stylist and the inherent properties of his working medium. The former are to some extent predictable but the latter errors are only really apparent to the stylist himself. There is no definable 'best' fit; rather the goodness of fit depends on human judgement. It is thus logical to use an interactive technique because no fully automatic technique can be expected to distinguish between the intent of the stylist and his errors and will at the best employ ad hoc procedures. Fig. 1 shows the kind of curve data which might be encountered in the car industry and acceptable and unacceptable (in the author's opinion) solutions. In order to be useful the interactive procedure must be easy to apply and a user should not need to know the mathematical principles involved. At Renault, data from a small clay model, or a hand sketched curve, is plotted, full size, on a drafting machine. The stylist then estimates graphically the parameters of an approximating curve which is then drawn by the machine. Three-dimensional curves are approximated in two plane projections. An acceptable approximation is usually achieved in a few iterations by adjusting the curve parameters. In some cases, no doubt, the stylist will do some redesigning as well as fitting. As smoothness is of paramount importance the system in effect gives the designer a 'perfect' medium in which to work rather than an imperfect one such as clay because the shapes which are created are basically smooth and irregularities must deliberately be designed. It is much easier to create a 'bump' in a curve than to remove an unintended 'bump' caused by bad data.