B\'ezier curves provide the basic building blocks of graphic design in 2D. In this paper, we port B\'ezier curves to manifolds. We support the interactive drawing and editing of B\'ezier splines on manifold meshes with millions of triangles, by relying on just repeated manifold averages. We show that direct extensions of the De Casteljau and Bernstein evaluation algorithms to the manifold setting are fragile, and prone to discontinuities when control polygons become large. Conversely, approaches based on subdivision are robust and can be implemented efficiently. We define B\'ezier curves on manifolds, by extending both the recursive De Casteljau bisection and a new open-uniform Lane-Riesenfeld subdivision scheme, which provide curves with different degrees of smoothness. For both schemes, we present algorithms for curve tracing, point evaluation, and point insertion. We test our algorithms for robustness and performance on all watertight, manifold, models from the Thingi10k repository, without any pre-processing and with random control points. For interactive editing, we port all the basic user interface interactions found in 2D tools directly to the mesh. We also support mapping complex SVG drawings to the mesh and their interactive editing.

中文翻译：

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b / Surf：曲面上的交互式贝塞尔样条曲线

B \'ezier曲线提供了2D图形设计的基本组成部分。在本文中，我们将B''ezier曲线移植到流形。我们仅依靠重复的流形平均值，就可以在具有数百万个三角形的流形网格上支持B \'ezier样条的交互式绘图和编辑。我们证明了De Casteljau和Bernstein评估算法对流形设置的直接扩展是脆弱的，并且当控制多边形变大时容易出现不连续性。相反，基于细分的方法是健壮的并且可以有效地实现。我们通过扩展递归De Casteljau二等分和新的开放均匀的Lane-Riesenfeld细分方案来定义流形上的B \'ezier曲线，该方案可提供具有不同平滑度的曲线。对于这两种方案，我们都会提供用于曲线跟踪的算法，点评估和点插入。我们对Thingi10k存储库中的所有防水，多头模型进行了鲁棒性和性能测试，无需任何预处理，并带有随机控制点。对于交互式编辑，我们将2D工具中发现的所有基本用户界面交互直接移植到网格上。我们还支持将复杂的SVG工程图映射到网格及其交互编辑。