Optimizing 4D Wires for Sparse 3D Abstraction

We present a unified framework for 3D geometric abstraction using a single continuous 4D wire, parameterized as a B-spline with spatial coordinates and variable width $(x,y,z,w)$. Existing approaches typically represent shapes as collections of many independent curve segments, which often leads to fragmented structures and limited physical realizability. In contrast, we show that a single continuous spline is sufficiently expressive to capture complex volumetric forms while enforcing global topological coherence. By imposing continuity, our method transforms 3D sketching from a local density-accumulation process into a global routing problem, providing a strong inductive bias toward cleaner aesthetics and improved structural coherence. To enable gradient-based optimization, we introduce a differentiable rendering pipeline that efficiently rasterizes variable-width curves with bounded projection error. This formulation supports robust optimization using modern guidance signals such as Score Distillation Sampling (SDS) or CLIP. We demonstrate applications including image-to-3D abstraction, multi-view wire art generation, and differentiable stylized surface filling. Experiments show that our unified representation produces structures with higher semantic fidelity and improved structural coherence compared to approaches based on collections of discrete curves.