On the Spline-Based Parameterisation of Plane Graphs via Harmonic Maps

TL;DR

This work develops a spline-based parameterisation framework for plane graphs that preserves interface conformity by parameterising each face with harmonic maps, enabling isogeometric analysis and dense mesh extraction. It combines a robust graph-preprocessing and templatisation stage with a harmonic-map-based face parameterisation, supported by a large offline catalogue of quadrangulation templates and multiple solution strategies. The method demonstrates bijective, conforming parameterisations on complex plane graphs (up to 42–55 faces after concave-corner handling), with post-processing steps to untangle folds and tune parametric properties. Overall, the approach provides a scalable, modular pipeline for multi-face parameterisation, suitable for PDE-based simulations and detailed mesh extraction, while highlighting areas for cross-face control and automation of concave-corner removal.

Abstract

This paper presents a spline-based parameterisation framework for plane graphs. The plane graph is characterised by a collection of curves forming closed loops that fence-off planar faces which have to be parameterised individually. Hereby, we focus on parameterisations that are conforming across the interfaces between the faces. Parameterising each face individually allows for the imposition of locally differing material parameters which has applications in various engineering disciplines, such as elasticity and heat transfer. For the parameterisation of the individual faces, we employ the concept of harmonic maps. The plane graph's spline-based parameterisation is suitable for numerical simulation based on isogeometric analysis or can be utilised to extract arbitrarily dense classical meshes. Application-specific features can be built into the geometry's mathematical description either on the spline level or in the mesh extraction step.

Figures (25)

• Figure 1: An example of a plane graph (left) that is visualised via its weight function $w(\, \cdot \,)$. The various polygons $\Omega_i$ that are bounded by the point sets $w(e), \, e \in F_i$ are highlighted in different colors. The graph contains several examples of two edges connecting the same two vertices, for example $e_{10}$ and $e_{13}$. The figure shows that $F_1 = \left(\pm e_1, \pm e_2, \ldots, \pm e_{12} \right)$. The right figure shows a possible spline-based parameterisation wherein each point set $w(e)$ is replaced by an accurate spline approximation $w^S(e)$ fencing off the spline domains $\Omega_i^S \approx \Omega_i$. Each face is parameterised from one or several patches using a harmonic map, while neighboring faces use the same spline edges $w^S(e)$, thus leading to a conforming interface.
• Figure 2: Figure showing various domains $\Omega$ (top) along with the associated parametric domain ${\hat{\Omega}}^{\mathbf{r}}$ we assign to each domain type (bottom). Dots indicate vertices that fence off the edges, wherein the root vertex is given in green while vertices that are modelled as a corner vertex are red and non-corner vertices are blue.
• Figure 3: Figure showing a template $\mathbb{T} \ni T_i = (V_i, E_i, \mathcal{Q}_i)$ for the $8$-sided regular polygon of radius one. The figure additionally shows the corresponding multipatch covering of a lens-shaped control domain ${\hat{\Omega}}^{\mathbf{r}}$ (see Section \ref{['sect:harmonic_maps']}). The control domain is parameterised by the controlmap $\mathbf{r}: {\hat{\Omega}}(8) \rightarrow {\hat{\Omega}}^{\mathbf{r}}$ that is constructed using optimisation of the inner patch vertices along with the Coons' patch approach. For details, see \ref{['sect:appendix_PAG']}.
• Figure 4: An example of removing a concave vertex (blue) via the use of Hermite splitting curves whose starting and end point tangents are contained in the endpoints' convex cones. The figure shows all possible Hermite curves originating in the bottom concave vertex $v \in V$. While all curves are valid, selection based on \ref{['eq:splitting_curve_quality_mu']} would favour the curve highlighted in blue since it is the straightest and removes two concave vertices at once.
• Figure 5: An example showing the splitting of a face with concave corners (blue) by drawing a Hermite curve from a concave corner (red) to another vertex (red or black). Algorithm \ref{['algo:concave_corners']} selects vertex pairs $\{ v_{\alpha}, v_{\beta} \} \subset \mathbb{V}(F)$, where $v_\alpha$ is a concave vertex in a greedy fashion and connects them using a Hermite curve, thus splitting the face in two. The same routine is then applied to the updated plane graph until no more eligible vertex pairs are found. In this example, the routine succeeds in fully automatically removing all concave vertices.

Theorems & Definitions (2)

• Theorem 1: Radó-Kneser-Choquet
• Remark