Chongyang Deng

Cages are compact geometric proxies widely used in computer graphics. Existing cage generation methods mainly focus on triangular cages. In this paper, we present the first automatic method for constructing high-quality quad-dominant tri-quad cages. Generating such cages is challenging because topology editing, geometric approximation, validity, and quadrilateral dominance are highly coupled in a constrained optimization problem. We make the problem tractable by decoupling topology and geometry while maintaining all cage-validity requirements throughout the pipeline, including being self-intersection-free, collision-free and feature-preserving. Starting from a feasible quad-dominant offset surface extracted from the signed distance field (SDF) of the input, we first perform constrained simplification. In this process, metric-aware and direction-preserving topology edits reduce complexity and suppress unnecessary triangles only when validity checks are satisfied. We then optimize the simplified feasible cage under the same validity constraints by minimizing a fidelity-and-quality energy that encourages tight approximation, rectangularity, smoothness, and orientation coherence. Because every update remains feasible and decreases the objective, validity is preserved by construction, and the algorithm converges to a stable local optimum. Comparative experiments demonstrate that our method produces tight, collision-free, and self-intersection-free quad-dominant tri-quad cages with high geometric fidelity and coherent directional structure. We further demonstrate their utility in nested cages construction, simplification, cage-based deformation, and subdivision-surface generation, showing advantages in approximation quality, deformation control, and directional coherence.

To address the interpolation problem of data points with normal constraints, we propose an interpolation algorithm for non-uniform B-spline curves (surfaces) that combines the augmented Lagrangian method with progressive-iterative approximation (PIA), which is termed AL-PIA (augmented Lagrangian-based progressive-iterative approximation).Firstly, the initial control points are obtained through parameterization and knot insertion of the given data points, so as to construct the initial sequence of non-uniform B-spline curves (surfaces). Then, in each iteration, an error vector satisfying interpolation conditions and energy minimization is constructed to update the control points iteratively. Finally, non-uniform B-spline curves (surfaces) interpolating the data points and satisfying normal constraints are generated. Theoretical analysis shows that the algorithm is convergent, and numerical experimental results verify the effectiveness of the AL-PIA algorithm. This method is applicable to planar curves, spatial curves and tensor product surfaces, providing a new approach with both theoretical completeness and computational efficiency for B-spline interpolation problems with geometric constraints.

We propose a novel automatic approach to construct low-complexity, manifold, and non-self-intersecting cages, which preserves geometric features, maintains tight enclosure, enforces symmetry, and produces high-quality meshes. The method is realized through a coherent three-stage pipeline. Starting with isosurface extraction from the Signed Distance Field (SDF), our method inherently enforces envelope and non-self-intersection properties via density field modification and leverages sharp features of the input to drive a shape-preserving initial cage. Second, a constrained Quadric Error Metric (QEM) simplification problem is solved using a geometry-driven push-pull strategy, decimating the initial cage to a user-specified vertex count. This yields a concise intermediate structure while preserving the fundamental properties. Finally, we introduce a symmetry-aware refinement stage via an integrated optimization that simultaneously preserves symmetry, improves mesh quality, and ensures tight alignment with the input. We evaluate symmetry using a deviation metric (SDE) to extract a symmetry plane and produce a symmetry-consistent cage, followed by mesh quality enhancement and collision-free enforcement. Through extensive experiments on diverse models, including evaluations on public benchmarks, we demonstrate that our method outperforms existing techniques in handling complex shapes, automatically preserving features and symmetry while generating a compact, coarse cage.

Transfinite interpolation is a fundamental theme in geometric modeling. The most well-known transfinite interpolation volume is the Coons volume. However, the topology of Coons volumes is restricted to hexahedra, which significantly limits their applications. In this paper, we generalize the Coons volume from hexahedral topology to arbitrary polyhedral topologies via generalized barycentric coordinates. We prove that the proposed generalized Coons volume possesses several desirable geometric properties and demonstrate its applications in computer graphics.

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.