Robust Error-Bounded Cage Generation for Planar Domains

Cages are crucial for generalized barycentric coordinates based shape editing and deformation applications. However, existing cage generation methods have no guarantee on the Hausdorff distance between the generated cage and the shape such that it is within a given distance bound. To address this, a robust planar cage generation algorithm is proposed. Firstly, given a planar shape and a Hausdorff distance bound, the algorithm generates an initial cage, ensuring that the Hausdorff distance between the planar shape and the cage satisfies the constraint. Secondly, the cage is simplified using topology optimization and geometric optimization while making sure that the distance constraint is always satisfied. Given any planar mesh as input, the algorithm can theoretically ensure that the generated cage satisfies the constraints. Through extensive experiments on a public planar shape dataset, the results demonstrate that this method can produce cages with fewer vertices given the same Hausdorff distance and smaller Hausdorff distance given the same number of vertices in comparison to the existing methods.