We consider B\'ezier curves with complex parameter, and we determine explicitly the affine iterated function system (IFS) corresponding to the de Casteljau subdivision algorithm, together with the complex parametric domain over which such an IFS has a unique global connected attractor. For a specific family of complex parameter having vanishing imaginary part, we prove that the Takagi fractal curve is the attractor, under suitable scaling.

翻译：我们研究了具有复参数的贝塞尔曲线，并显式地给出了与德卡斯特拉奥细分算法相对应的仿射迭代函数系统（IFS），同时确定了该IFS具有唯一全局连通吸引子的复参数域。针对虚部为零的一类特殊复参数族，我们证明了在适当缩放条件下，高木分形曲线即为该吸引子。