Rotationally Seamless Conformal Parameterization

• Rotationally seamless conformal parameterization is a technique that constructs global conformal maps with discrete rotation transitions (e.g., multiples of π/2) ensuring consistent holonomy signatures.
• The methodology employs strategies like Penner coordinates, mixed-integer optimization, and harmonic padding to manage cone singularities and quantized angle constraints efficiently.
• Applications span quad meshing, texture mapping, and vector-field-aligned remeshing, providing robust and practical solutions in computational geometry.

Rotationally seamless conformal parameterization is a fundamental computational operation in geometry processing, enabling the construction of piecewise-conformal atlases on two-dimensional manifolds in which all chart transitions are rigid rotations by multiples of $π/2$ or, more generally, arbitrary discrete angles. This property is essential for downstream applications such as quad meshing, texture synthesis, and vector-field-aligned parameterizations, where consistency and controllable holonomy signatures are required. The theoretical and algorithmic developments in this area focus on producing globally conformal surface mappings with precisely prescribed singularities (cones) and rotational transitions along homology generators, subject to quantized angle and holonomy constraints.

1. Mathematical Foundations and Holonomy Signatures

At the core of rotationally seamless conformal parameterization is the quantized control of cone singularities and holonomy. Consider a triangulated surface $M$ of genus $g$, with vertex set $V$, edge set $E$, and face set $F$. A discrete metric assigns positive edge-lengths $\ell:E\to\mathbb{R}_+$, inducing flat Euclidean triangles with curvature concentrated at vertices ("cones"), quantified by $K_i(\ell)=2\pi-\sum_{T\ni i}\alpha_i^T(\ell)$. Holonomy signatures consist of integer multiples of $\pi/2$ prescribed at cone singularities and along $2g$ fundamental homology loops (dual cycles)

$$\sum_{T\ni c}\alpha_c^T = 2\pi - \kappa_c, \quad \kappa_c\in\{\pm\frac{\pi}{2}, \pm\pi, \dots\},$$

$$\sum_{m=1}^{n_j}d_m^j \alpha_m^j = k^\ell_j \frac{\pi}{2}, \quad k^\ell_j\in\mathbb{Z}.$$

The total holonomy signature must satisfy Gauss–Bonnet-type conditions (e.g., $\sum_i k^v_i + \sum_{j=1}^{2g}k^\ell_j = 4(1-g)$ for closed orientable surfaces), ensuring global consistency (Capouellez et al., 31 Jul 2024).

2. Algorithmic Approaches and Optimization Frameworks

Modern computational strategies formulate the problem as an underdetermined system of nonlinear or mixed-integer constraints on vertex angle sums and loop holonomy, typically in the space of metrics or vertex scaling factors:

• Penner Coordinate Approach: The "Seamless Parametrization in Penner Coordinates" algorithm (Capouellez et al., 31 Jul 2024) operates in the global space of log-edge lengths ("Penner coordinates") on a fixed triangulation. At each step, the triangulation is flipped to intrinsic Delaunay, and the system $$F(\lambda)=C\alpha(\mathrm{Del}(M_0,\lambda))-\Theta=0$$ is solved by extended Newton iteration:

$$d = J^T(JJ^T)^+(-F), \text{ with backtracking to ensure descent.}$$

Intrinsic edge flips and accumulated Jacobians align with convex optimization perspectives.

• Mixed-Integer Cone Optimization: The algorithm of (Du et al., 24 Dec 2025) reduces the problem to small-scale mixed-integer quadratic programs (MIQPs) over cone angles and positions. By alternately optimizing cone locations, integer Quantized angles (multiples of $\pi/2$), and counts, the method efficiently minimizes distortion functional (e.g., squared Hencky energy) under discrete Yamabe flow-type linear constraints and integer quantization of singular curvature.
• Holonomy-Constrained Conformal Maps: Campen & Zorin (Campen et al., 2017) use vertex-based log-conformal scaling $u:V\to\mathbb{R}$ and minimize a convex energy whose gradient yields per-vertex angle and loop holonomy constraints. Newton's method with a line search ensures global convergence within the space where triangles remain valid (non-degenerate), supporting arbitrary prescribed holonomy.
• Seamlessness Modification by Boundary Padding: The constructive existence result of (Campen et al., 2018) employs convex harmonic map computation on a cut surface, followed by a linear seam-equalization procedure ("padding") to enforce exact matching (to within a prescribed rotation) across chart boundaries. The resulting system of boundary unknowns is always solvable for generic admissible cone data, leading to a fully seamless, locally-injective atlas.

3. Ensuring Rotational Seamlessness and Conformality

Rotational seamlessness requires that on identification of any pair of cut-edges or segments, the corresponding charts in the atlas are related by a rigid rotation of $k\times \pi/2$, where $k$ is an integer holonomy signature. This is achieved by:

• Simultaneously enforcing per-vertex angle sums and per-loop holonomy to quantized multiples.
• In the Penner coordinates or discrete conformal map frameworks, embedding the surface in the plane is locally isometric—hence conformal—inside each triangle. Chart transitions are isometries (rotations of $π/2$ multiples), yielding true rotationally seamless atlases (Capouellez et al., 31 Jul 2024, Campen et al., 2017, Campen et al., 2018).
• In genus $g$ surfaces, holonomy constraints around the $2g$ fundamental cycles are explicitly encoded as part of the system, with additional variables or constraints as needed (Du et al., 24 Dec 2025).

4. Performance, Practical Implementation, and Complexity

The efficiency and scalability of rotationally seamless parameterization algorithms are demonstrated on large-scale, high-genus datasets:

• The Penner coordinate solver converges robustly in approximately $9-12$ iterations for up to $4,307$ genus and meshes with up to $8\times 10^5$ vertices, with RMS log-length distortions typically $10^{-3}$–$10^{-2}$ (Capouellez et al., 31 Jul 2024).
• Mixed-integer methods achieve average runtimes of $1$–$10$ seconds on genus-0 meshes of $10^4$ vertices (mean cone count $24\pm 7$), with up to $30\times$ speedup compared to previous approaches (Du et al., 24 Dec 2025).
• The harmonic/padding pipeline of (Campen et al., 2018) solves all convex subproblems and final seam-equalization in linear or nearly-linear time, with practical cone configurations always admitted except for the unique inadmissible sphere $(3,5)$-cone case.

A summary comparison:

Method	Holonomy Control	Metric Embedding	Main Solver	Average Iterations / Time
Penner coordinates (Capouellez et al., 31 Jul 2024)	Vertex + Loops	Yes	Newton/Cholesky	9–12 / 0.01–0.1s
MIQP cones (Du et al., 24 Dec 2025)	Vertex + Loops	Yes	MIQP + Poisson	4.5s avg (10k verts)
Holonomy scale (Campen et al., 2017)	Vertex + Loops	No (similarity)	Newton	O(10) iterations
Harmonic+padding (Campen et al., 2018)	Vertex only	Yes	Convex QP + LinSys	~0.02s-1s

5. Theoretical Guarantees, Existence, and Limitations

The existence of rotationally seamless conformal parameterizations is guaranteed whenever the holonomy signature meets Gauss–Bonnet constraints and avoids known exceptional cases (specifically the $(3,5)$-cone prescription on the genus-0 sphere) (Campen et al., 2018). Convexity underlies both Newton-type and some padding-based solvers; the global convergence for the full holonomy-constrained Penner coordinate approach is empirically observed but not yet formally proved (Capouellez et al., 31 Jul 2024). For the mixed-integer formulation, exact MIQP solutions are tractable only for limited cone counts, motivating local pruning and alternation strategies for scalability (Du et al., 24 Dec 2025).

Limitations include sensitivity to extremely poor initial triangulation quality (requiring intrinsic refinement or interpolation for robustness), current restriction to closed surfaces with boundary extensions via doubling or prescribed holonomy, and lack of guaranteed injectivity for certain extreme configurations.

6. Applications and Relationship to Related Methodologies

Rotationally seamless conformal parameterizations are foundational in quad mesh generation, construction of global UV atlases for texture mapping, registration, and field-aligned remeshing. The ability to enforce and control holonomy (both cone angles and chart transition angles) is critical for mesh layouts that, for instance, align with principal curvature or feature directions or for procedural texture synthesis. The formalism relates closely to discrete uniformization, Ricci-flow, cone metric geometry, and the theory of Penner coordinates for decorated Teichmüller space.

Comparison against prior approaches reveals that:

• Traditional scale-factor conformal parameterizations can only prescribe vertex-angle constraints and are insufficient for arbitrary holonomy signature control (Capouellez et al., 31 Jul 2024).
• Similarity-optimized frameworks produce rotationally seamless surface embeddings but do not guarantee metric realization or injectivity.
• The harmonic and padding-based existence framework gives constructive proof and an efficient pipeline for arbitrary admissible signatures, robustly producing locally-injective, rotationally seamless maps.

7. Outlook and Future Directions

Open problems include rigorous proof of global convergence for Penner-coordinate-based holonomy solvers, extension to surfaces with boundaries and sharp feature constraints, further reduction of metric distortion especially under sparse cone distributions, and tighter integration of parameterization algorithms with globally injective and distortion-limited surface flattening techniques.

The rapid progress in efficient, robust, and general algorithms for rotationally seamless conformal parameterization continues to underpin advances across computational geometry, meshing, and digital fabrication workflows (Capouellez et al., 31 Jul 2024, Du et al., 24 Dec 2025, Campen et al., 2018).