In this paper we develop the formalism of rational complex Bezier curves. This framework is a simple extension of the CAD paradigm, since it describes arc of curves in terms of control polygons and weights, which are extended to complex values. One of the major advantages of this extension is that we may make use of two different groups of projective transformations. Besides the group of projective transformations of the real plane, we have the group of complex projective transformations. This allows us to apply useful transformations like the geometric inversion to curves in design. In addition to this, the use of the complex formulation allows to lower the degree of the curves in some cases. This can be checked using the resultant of two polynomials and provides a simple formula for determining whether a rational cubic curve is a conic or not. Examples of application of the formalism to classical curves are included.

翻译：本文发展了有理复贝塞尔曲线的形式化理论。该框架是计算机辅助设计（CAD）范式的简单扩展，因为它用控制多边形和权重来描述曲线弧段，这些参数被扩展至复数值。此扩展的主要优势之一在于我们可以利用两类不同的射影变换群：除了实平面的射影变换群外，我们还拥有复射影变换群。这使得我们能够将几何反演等实用变换应用于设计中的曲线。此外，在某些情况下，采用复数表述可以降低曲线的次数。这可通过计算两个多项式的结式进行验证，并为判断有理三次曲线是否为圆锥曲线提供了简洁公式。文中包含了该形式化理论在经典曲线中的应用实例。