贝塞尔曲线

Prior to UNISURF, designing the complex, flowing surfaces of a car body was a manual, labor-intensive process involving physical clay models and templates. Pierre Bézier’s work at Renault aimed to translate this physical design process into a mathematical and computational framework. The result was UNISURF (Unification of Surfaces), a system that allowed designers to define and manipulate freeform surfaces on a computer. The mathematical foundation of this system was the Bézier curve. A Bézier curve is a parametric curve defined by a set of control points. For a cubic Bézier curve, four points are used: two endpoints that the curve passes through, and two intermediate control points that define the curve’s shape and tangent directions. The curve itself does not typically pass through these intermediate points, but they act as ‘handles’ that designers can intuitively manipulate to sculpt the curve’s shape.

This concept was extended to surfaces, creating Bézier surfaces (or patches) defined by a grid of control points. By stitching these patches together with specific continuity conditions (e.g., G0 for position, G1 for tangency), complex and smooth surfaces like a car’s hood or fender could be modeled precisely. The mathematical representation is a polynomial function, for a cubic Bézier curve it is [latex]B(t) = (1-t)^3 P_0 + 3(1-t)^2 t P_1 + 3(1-t) t^2 P_2 + t^3 P_3[/latex], for [latex]t in [0, 1][/latex]. This mathematical rigor allowed the design data to be used directly for manufacturing (CAM), such as programming CNC milling machines to create dies. This tight integration of design and manufacturing was a hallmark of UNISURF and a major step forward for industrial production.