The holonomy angle of a dual loop is the sum of signed inner angles along the strip — equivalently, the angle between first and last edge after laying the strip flat.

Seamless parametrization

A piecewise-linear, locally injective map $F:M^c\to\mathbb{R}^2$ such that for every cut edge $e$ with mate $e'$, there is a rigid transform $\sigma_e(x)=R_e x + t_e$ with $R_e$ a rotation by an integer multiple of $\pi/2$, mapping $F(e)$ to $F(e')$.

Holonomy angle of a loop $\gamma$

and the holonomy number is $k_\gamma^F = \kappa_\gamma^F / (2\pi)$. For seamless maps it lives in $\tfrac14\mathbb{Z}$.

Cone index

For a vertex loop $\partial v^*$, the index is $I_v^F = 1 - k_{\partial v^*}^F \in \tfrac14\mathbb{Z}$.

Definition

Holonomy numbers along a homology basis of $M\setminus C$:

This finite signature determines the holonomy number of every loop on the surface (via Gauss-Bonnet).

Equivalence (non-uniqueness)

Different choices of basis loops give different numbers but represent the same parametrization topology — see right.

Connection to cross fields

For a discrete cross field with turning numbers $T_\gamma$ on a homology basis, matching $T$ to $k$ on the basis forces a match for every loop. This is a Poincaré-Hopf-style identity:

Cross fields with any prescribed turning-number signature can be built by a linear solve [Crane et al., Trivial Connections]. Our method extends this topological control to parametrizations.

Two equivalent signatures. Cones: red $-\tfrac14$, blue $+\tfrac14$. Loop holonomy numbers boxed (0 vs $\tfrac14$). Sliding the loop across the leftmost red cone changes the number by exactly $-I_v = +\tfrac14$.

The question

For which holonomy signatures does a seamless parametrization exist?

Our answer (Prop. 2)

Let $H=\{\gamma_1,\dots,\gamma_{2g}\}$ be a system of loops cutting $M$ to a disk, and let cones $v_1,\dots,v_m$ have prescribed indices $I_1,\dots,I_m\in\tfrac14\mathbb{Z}$. If

then for any target shifts $k_1,\dots,k_{2g}\in\tfrac14\mathbb{Z}$, there is an alternative basis $\delta_1,\dots,\delta_{2g}$ (still cutting $M$ to a disk) with

Cross-field topology and seamless-map topology nearly coincide — the gap is essentially empty in practice.

Two non-intersecting loops $\gamma,\delta$ joined by a path $\alpha$.

Rerouting a loop around a cone of index $\tfrac14$ twice — holonomy changes by $\tfrac12$.

Quasi-Additivity (Prop. 1)

If $\alpha$ connects the right side of $\gamma$ to the right side of $\delta$, there is a simple loop $\gamma_0$ — arbitrarily close to $\gamma\cup\alpha\cup\delta$ — with

If $\alpha$ goes left-to-left, the $-1$ becomes $+1$.

Specialisation: rerouting around a cone $v_i$

Take $\delta = \partial v_i^*$ (the cone loop, $k_\delta = 1 - I_{v_i}$). Then

Sign depends on which side of $\gamma$ the path $\alpha$ comes in from.

Repeated rerouting around chosen cones can shift a loop's holonomy number by any integer combination of cone indices — an Extended Euclidean Algorithm reaches any target if the GCD condition holds.

A hole-chain cut graph. Highlighted in red: a loop making two left turns — its parametrization holonomy number will be $\tfrac24 = \tfrac12$.

Hole-chain cut graph

Built from $g$ non-contractible loops (one per handle) plus $2g-1$ connector paths chaining them. Both are computed on a four-sheeted cover $M_4$ using a cross-field-aligned anisotropic metric [Diaz et al. 2009; Campen 2014].

Why field-guided?

Soft-guidance promotes loops whose left/right-turn balance already matches the field's turning numbers — minimising downstream rerouting.

Homology basis $H$

Take a spanning tree $T$ of $G$. Each of the remaining $2g$ bridge edges defines a unique loop (bridge + tree paths to root). These $2g$ loops form a system of loops.

Crucial fact: each bridge belongs to exactly one basis loop, so we can shift each loop's holonomy independently.

SP holonomy pattern

Along any loop in $G$, the SP method's output has holonomy number $\tfrac14(m_\ell - m_r)$ (left turns minus right turns). Our job is to make this balance equal the target.

For each basis loop with current balance $\tfrac14(m_\ell - m_r)$ and target $t$, reroute its bridge segment so that the balance changes by

Tier 1 — Holonomy-aware Dijkstra

• Dual shortest path on $M^c$, with edges tagged by the index sum of cones they enclose w.r.t. the bridge segment.
• Terminates as soon as the enclosed index sum reaches the required holonomy shift.
• Field-aware costs → usually returns a clean, simple replacement path.
• May fail (path turns out non-simple after projection) — fall through.

Tier 2 — Constructive fallback

• Pick a multiset $V$ of cones with index sum $k$ — found by the Extended Euclidean Algorithm under the GCD condition.
• For each $v^*\in V$, splice the bridge tightly around $v^*$ via the shortest path on the appropriate side.
• Edge splits ensure the new path stays disjoint from the rest of $G$.
• Provably succeeds whenever the GCD condition holds.

Two different rerouted loop systems for the same prescribed holonomy. The final parametrizations are identical up to a seamless transformation.

Cut-aligned conformal metric

On the cut mesh $M^c$ we compute a discrete conformal metric with

• prescribed cone angles at vertices of $C$,
• prescribed geodesic curvature along the cut $G'$ — segments become straight, branch corners become right angles.

We use the convergent algorithm of Campen et al. 2021 / Gillespie 2021 (the same hyperbolic-Delaunay framework presented in our SIGGRAPH Asia 2021 paper), which gives strict convergence guarantees on these challenging boundary shapes.

Padding

Mate boundary segments are straight and meet at multiples of $\pi/2$, but lengths may differ. Stretch a thin cone-free strip along each segment so that mated segments equalise — provably feasible, preserves seamlessness.

Final optimisation

Projected Newton on cross-field alignment $E_A$ + small symmetric Dirichlet $E_D$ as injectivity barrier; reduced-row-echelon parameterisation of constraints for stability.

Padding in the parameter domain: a thin strip along the top straight cut segment is stretched vertically; vertices then shift horizontally to match their mates across the cut.

A genus-2 model walked through the rerouting + padding pipeline.

Empirical observations

• Tested on the standard Myles 2014 dataset (genus $> 0$).
• Cut-graph stage succeeds in 100% of cases (all satisfy the GCD condition; everything is combinatorial and exact).
• Field-guided Dijkstra resolves the majority of reroutings; constructive fallback is invoked only occasionally.
• Final parametrization succeeds in most cases — a handful of high-genus models hit numerical limits in optimisation, not topology.