We present a method to construct quadratic birational planar maps, which is based on the observation that any rational planar map with complex rational representation possesses two special syzygies. After establishing the relation between degree one complex rational Bézier curves and quadratic rational Bézier curves, we derive conditions to determine when a quadratic rational planar map has a complex rational representation. Hence, we can make a quadratic planar map be birational by suitably choosing the middle Bézier control points and their corresponding weights. In addition, this paper explores an interesting geometric property of the isoparametric curves of quadratic birational planar maps with complex rational representation, i.e., all isoparametric curves meet at a common point.

我们提出了一种构造二次双平平面图的方法，该方法基于以下观察：具有复杂有理表示形式的任何有理平面图都具有两个特殊的syzysy。在建立一阶复杂有理Bézier曲线和二次有理Bézier曲线之间的关系之后，我们得出确定二次有理平面图何时具有复杂有理表示的条件。因此，通过适当地选择中间的贝塞尔控制点及其相应的权重，我们可以使二次平面图成为双边的。此外，本文探索了具有复杂有理表示的二次双边平面图的等参曲线的有趣几何性质，即所有等参曲线在一个公共点处交汇。