We present Advancing Front Mapping (AFM), a provably robust algorithm for the computation of surface mappings to simple base domains. Given an input mesh and a convex or star-shaped target domain, AFM installs a (possibly refined) version of the input connectivity into the target shape, generating a piece-wise linear mapping between them. The algorithm is inspired by the advancing front meshing paradigm, which is revisited to operate on two embeddings at once, thus becoming a tool for compatible mesh generation. AFM extends the capabilities of existing robust approaches, such as Tutte or Progressive Embedding, by providing the same theoretical guarantees of injectivity and at the same time introducing two key advantages: support for a broader set of target domains (star-shaped polygons) and local mesh refinement, which is used to automatically open the space of solutions in case a valid mapping to the target domain does not exist. AFM relies solely on two topological operators (split and flip), and on the computation of segment intersections, thus permitting to compute provably injective mappings without solving any numerical problem. This makes the algorithm predictable, easy to implement, debug and deploy. We validated the capabilities of AFM extensively, executing more than one billion advancing front moves on 36K mapping tasks, proving that our theoretical guarantees nicely transition to a robust and practical implementation.

翻译：我们提出了前沿推进映射（AFM），一种用于将曲面映射到简单基域的鲁棒性算法。给定输入网格和凸形或星形目标域，AFM 将输入连接性（可能经过细化）安装到目标形状中，生成它们之间的分片线性映射。该算法受前沿推进网格划分范式的启发，但重新设计为同时作用于两个嵌入，从而成为一种兼容网格生成的工具。AFM 扩展了现有鲁棒方法（如 Tutte 嵌入或渐进嵌入）的能力，提供了相同的单射性理论保证，同时引入两个关键优势：支持更广泛的目标域（星形多边形）和局部网格细化——当有效映射到目标域不存在时，后者用于自动打开解空间。AFM 仅依赖两种拓扑操作（分裂和翻转）以及线段交集计算，从而无需解决任何数值问题即可计算可证明的单射映射。这使得算法具有可预测性，易于实现、调试和部署。我们广泛验证了 AFM 的能力，在 36K 个映射任务上执行了超过十亿次前沿推进移动，证明了我们的理论保证能够良好地转化为鲁棒且实用的实现。