Cages are fundamental structures in computer graphics, serving as versatile proxies for a wide range of applications. A high-quality cage must balance two competing objectives: minimizing the face count to ensure simplicity, and maximizing tightness to maintain high geometric fidelity to the input mesh. In this paper, we propose PR-Cage, a nested optimization framework for automated cage generation. For the outer control layer, we introduce a thickness parameter τ that defines a feasibility region; the evolving cage is guided by the τ -offset surface. We observe that an optimal balance between simplicity and tightness is achievable by progressively relaxing the parameter τ via a staircase schedule. For the inner iterations, we extend the traditional Quadric Error Metric (QEM) framework by incorporating rigorous linear inequality constraints to suppress triangle degeneration and prevent normal flips. Our algorithm relies exclusively on the atomic operations of edge collapses and edge flips, resulting in high computational efficiency and robustness. Comparative experiments on public datasets demonstrate that PR-Cage consistently outperforms existing methods, achieving extreme simplification while maintaining high adherence to the underlying geometry; see the teaser figure. Due to these favorable properties, we demonstrate the utility of our method in several downstream applications, such as contact simulation and deformation, where PR-Cage exhibits significant advantages in both quality and performance.

Space deformations deform the ambient space and thus implicitly deform the embedded objects. Free-Form Deformation allows high-order deformation to the embedding space, yet the lattice may fail to conform to the object and involves many internal control points. Cage-based Deformation utilizes a cage space that conforms to the object, obviating the need for additional internal control points, but it is typically linear at edges. In this paper, we propose a simple and general method with both advantages while avoiding their drawbacks, allowing users to implement high-order cage-based deformation. To achieve this goal, we introduce a new parametric transfinite interpolation scheme based on generalized barycentric coordinates, which unifies and generalizes the rectangular and triangular Coons patch. This C 0 Generalized Coons patch can be defined not only over 2D domains but also 3D domains or even higher-dimensional domains, with arbitrary polytopes, even including non-manifold topologies. Moreover, the C 0 Generalized Coons patch has an elegant mathematical expression.