In Farin (2006) Farin proposed a method for designing Bézier curves with monotonic curvature and torsion. Such curves are relevant in design due to their aesthetic shape. The method relies on applying a matrix $M$ to the first edge of the control polygon of the curve in order to obtain by iteration the remaining edges. With this method, sufficient conditions on the matrix $M$ are provided, which lead to the definition of Class A curves, generalizing a previous result by Mineur et al. (1998) for plane curves with $M$ being the composition of a dilatation and a rotation. However, Cao and Wang (2008) have shown counterexamples for such conditions. In this paper, we revisit Farin’s idea of using the subdivision algorithm to relate the curvature at every point of the curve to the curvature at the initial point in order to produce a closed formula for the curvature of planar curves in terms of the eigenvalues of the matrix $M$ and the seed vector for the curve, the first edge of the control polygon. Moreover, we give new conditions in order to produce planar curves with monotonic curvature. The main difference is that we do not require our conditions on the eigenvalues to be preserved under subdivision of the curve. This facilitates giving a unified derivation of the existing results and obtain more general results in the planar case.

在Farin（2006）中，Farin提出了一种设计具有单调曲率和扭转力的Bézier曲线的方法。这样的曲线由于其美学形状而在设计上是相关的。该方法依赖于应用矩阵$\mathrm{中号}$到曲线的控制多边形的第一边缘，以便通过迭代获得剩余的边缘。使用这种方法，矩阵上有足够的条件$\mathrm{中号}$提供了这些信息，从而定义了A类曲线，从而概括了Mineur等人先前的结果。（1998）对于具有$\mathrm{中号}$是膨胀和旋转的组成。但是，曹和王（2008）给出了这种情况的反例。在本文中，我们将重新讨论Farin的思想，即使用细分算法将曲线的每个点的曲率与初始点的曲率相关联，以便根据平面的特征值生成平面曲线曲率的封闭公式。矩阵$\mathrm{中号}$以及曲线的种子向量，即控制多边形的第一条边。而且，我们给出了新的条件以便产生具有单调曲率的平面曲线。主要区别在于，我们不需要将特征值的条件保留在曲线的细分范围内。这有助于给出现有结果的统一推导，并在平面情况下获得更一般的结果。