The problem of seamless parametrization of surfaces is of interest in the context of structured quadrilateral mesh generation and spline‐based surface approximation. It has been tackled by a variety of approaches, commonly relying on continuous numerical optimization to ultimately obtain suitable parameter domains. We present a general combinatorial seamless parameter domain construction, free from the potential numerical issues inherent to continuous optimization techniques in practice. The domains are constructed as abstract polygonal complexes which can be embedded in a discrete planar grid space, as unions of unit squares. We ensure that the domain structure matches any prescribed parametrization singularities (cones) and satisfies seamlessness conditions. Surfaces of arbitrary genus are supported. Once a domain suitable for a given surface is constructed, a seamless and locally injective parametrization over this domain can be obtained using existing planar disk mapping techniques, making recourse to Tutte's classical embedding theorem.

中文翻译：

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无缝参数域的组合构造

在结构化四边形网格生成和基于样条的曲面近似的背景下，曲面的无缝参数化问题很有趣。它已通过多种方法解决，通常依靠连续数值优化来最终获得合适的参数域。我们提出了一个通用的组合无缝参数域构造，在实践中没有连续优化技术固有的潜在数值问题。这些域被构造为抽象的多边形复合体，可以嵌入到离散平面网格空间中，作为单位正方形的联合。我们确保域结构匹配任何规定的参数化奇点（锥体）并满足无缝条件。支持任意属的表面。