Conformal tubular parameterization and toroidal bending of tube-like surfaces

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.