Tight Quad-Dominant Tri-Quad Cage Construction for 3D Shapes

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.