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.

Abstract Any point inside a d‐dimensional simplex can be expressed in a unique way as a convex combination of the simplex's vertices, and the coefficients of this combination are called the barycentric coordinates of the point. The idea of barycentric coordinates extends to general polytopes with n vertices, but they are no longer unique if n > d+1. Several constructions of such generalized barycentric coordinates have been proposed, in particular for polygons and polyhedra, but most approaches cannot guarantee the non‐negativity of the coordinates, which is important for applications like image warping and mesh deformation. We present a novel construction of non‐negative and smooth generalized barycentric coordinates for arbitrary simple polygons, which extends to higher dimensions and can include isolated interior points. Our approach is inspired by maximum entropy coordinates, as it also uses a statistical model to define coordinates for convex polygons, but our generalization to non‐convex shapes is different and based instead on the project‐and‐smooth idea of iterative coordinates. We show that our coordinates and their gradients can be evaluated efficiently and provide several examples that illustrate their advantages over previous constructions.

Abstract Pseudo‐splines provide a rich family of subdivision schemes with a wide range of choices that meet various demands for balancing the approximation power, the length of the support, and the regularity of the limit functions. Special cases of pseudo‐splines include uniform odd‐degree B‐splines and the interpolatory 2n‐point subdivision schemes, and the other pseudo‐splines fill the gap between these two families. In this paper we show how the refinement step of a pseudo‐spline subdivision scheme can be implemented efficiently using repeated local operations, which require only the data in the direct neighbourhood of each vertex, and how to generalize this concept to quadrilateral meshes with arbitrary topology. The resulting pseudo‐spline surfaces can be arbitrarily smooth in regular mesh regions and C1at extraordinary vertices as our numerical analysis reveals.

For approximating subdivision schemes, there are several unified frameworks for effectively constructing subdivision surfaces generalizing splines of an arbitrary degree. In this article, we present a similar unified framework for interpolatory subdivision schemes. We first decompose the 2 n -point interpolatory curve subdivision scheme into repeated local operations. By extending the repeated local operations to quadrilateral meshes, an efficient algorithm can be further derived for interpolatory surface subdivision. Depending on the number n of repeated local operations, the continuity of the limit curve or surface can be of an arbitrary order C L , except in the surface case at a limited number of extraordinary vertices where C 1 continuity with bounded curvature is obtained. Boundary rules built upon repeated local operations are also presented.

This paper presents an interpolatory subdivision scheme derived from the Doo-Sabin subdivision scheme. We first present the relations among three curve subdivision schemes, namely a four point interpolatory subdivision scheme, a cubic B-spline curve subdivision scheme, and the Chaikin's algorithm that generates uniform quadratic B-spline curves. By generalizing these relations to the surface case, we derive an interpolatory surface subdivision scheme from the Doo-Sabin subdivision scheme, a generalization of the Chaikin's algorithm to surface subdivision. In the new subdivision scheme, we also introduce a variable tension parameter that is dependent to local control vertices. The variable tension parameter can be used to effectively control the resulting limit surface of the proposed subdivision scheme.

Interpolating an arbitrary topology mesh by a smooth surface plays an important role in geometric modeling and computer graphics. In this paper we present an efficient new algorithm for constructing a Doo-Sabin subdivision surface that interpolates a given mesh. By introducing additional degrees of freedom, the control vertices of the Doo-Sabin subdivision surface can be obtained directly with no need to solve any initial or intermediate large systems. The control points are computed by modifying the geometric rules of the first step of Doo-Sabin subdivision scheme and the resulting surface interpolates given vertices and optionally normal vectors at the vertices. The method has several merits for surface modeling purposes: (1) Efficiency: we obtain a generalized quadratic B-spline surface to interpolate a given mesh in a robust and simple manner. (2) Simplicity: we use only simple geometric rules to construct a smooth surface interpolating given data. (3) Locality: the perturbation of a given vertex only influences the surface shape near this vertex. (4) Freedom: for each vertex, there is one degree of freedom to adjust the shape of the interpolation surface. These features make surface interpolation using Doo-Sabin surface very simple and thus make the method itself suitable for interactive free-form shape design.