In this paper, we introduce a new averaging rule, the nonlinear weighted averaging rule. As an application, this averaging rule is used to replace the midpoint averaging in the de Casteljau evaluation algorithm and with this scheme we can also generate transcendental functions which cannot be generated by the classical de Casteljau algorithm. We also investigate the properties of the curves of the functions generated by blossoming, where the results show that these curves and the classical Bézier curves have some similar properties, including variation diminishing property and endpoint interpolation. However, the curves obtained by blossoming using nonlinear weighted averaging rules induced by certain functions violate some properties like convex hull property.

中文翻译：

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非线性加权平均与开花

在本文中，我们介绍了一种新的平均规则，即非线性加权平均规则。作为一种应用程序，该求平均规则用于代替de Casteljau评估算法中的中点求平均值，并且使用此方案，我们还可以生成经典de Casteljau算法无法生成的超越函数。我们还研究了由开花产生的函数的曲线的特性，结果表明这些曲线和经典的贝塞尔曲线具有一些相似的特性，包括变化减小特性和端点插值。但是，通过使用由某些函数引起的非线性加权平均规则进行开花而获得的曲线违反了某些性质，例如凸包性质。