Tube-like surfaces are widely encountered in geometry processing, engineering structures, and medical anatomy, yet their intrinsic longitudinal and circumferential topology is not well preserved by conventional planar annular or rectangular parameterization domains. In this work, we propose a conformal parameterization framework for open tube-like surfaces with two boundary components. The proposed method first constructs a fixed-boundary tubular parameterization by cutting the input mesh, computing a disk-to-rectangle conformal map, and lifting the result to a three-dimensional tubular domain. To reduce residual distortion introduced near the cut seam, we further introduce a localized quasi-conformal correction scheme formulated on an annular domain, which improves conformality while leaving regions away from the seam unchanged. To handle noisy or irregular input boundaries, we also develop a free-boundary variant based on boundary extension and cycle-Laplacian smoothing, allowing the prescribed boundary constraints to be imposed on artificial outer rings rather than directly on the original surface. Finally, we derive two conformal toroidal bending maps that transform the tubular parameterization into toroidal geometries while preserving the underlying tube topology. Experiments on synthetic tube meshes and real vascular surfaces demonstrate that the proposed framework produces low-distortion parameterizations, effectively mitigates seam-induced artifacts, improves robustness for boundary-noisy inputs, and provides flexible tubular and toroidal target domains for downstream surface processing tasks.

翻译：管状曲面在几何处理、工程结构及医学解剖中广泛存在，然而传统平面环形或矩形参数化域难以良好保留其固有的纵向与环向拓扑结构。本文针对具有两个边界的开管状曲面，提出一种保形参数化框架。该方法首先通过切割输入网格、计算圆盘到矩形的保形映射并将结果提升至三维管状域，构建固定边界的管状参数化。为减少切割缝附近引入的残余形变，我们进一步引入基于环形域建立的局部拟保形修正方案，在保持远离切割缝区域不变的同时提升共形性。针对含噪声或不规则输入边界，我们基于边界延拓和循环拉普拉斯平滑开发了自由边界变体，使得预设边界约束可施加于人工外环而非原始曲面表面。最后，我们推导出两种保形环形弯曲映射，在保持底层管状拓扑的同时将管状参数化转换为环形几何。在合成管状网格与真实血管表面上的实验表明，所提框架能够产生低形变参数化，有效缓解切割缝伪影，提升边界噪声输入的鲁棒性，并为下游曲面处理任务提供灵活的管状与环形目标域。