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：我们提出了一种鲁棒性可证明的算法——推进前沿映射（AFM），用于计算曲面到简单基域的映射。给定输入网格和凸或星形目标域，AFM将（可能经细化的）输入连通性安装到目标形状中，生成两者之间的分段线性映射。该算法受推进前沿网格划分范式启发，通过同时在两种嵌入上运行，将其转变为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，在提供相同单射性理论保证的同时，引入两个关键优势：支持更广泛的目标域（星形多边形）以及局部网格细化——当到目标域的有效映射不存在时，该机制自动打开解空间。AFM仅依赖两种拓扑算子（分裂与翻转）以及线段交点的计算，从而无需求解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。我们通过超过10亿次推进前沿移动操作在36,000个映射任务上广泛验证了AFM的能力，证明了我们的理论保证能够完美过渡到鲁棒且实用的实现中。