Vertex-Sharing Mesh Parameterization

• The paper introduces a method that maps C² planar curves onto triangulated domains using closest point projection, ensuring a global homeomorphism with C¹ regularity on each edge.
• It employs geometric criteria such as acute conditioning angles and curvature-based mesh quality measures to maintain accurate Jacobian bounds and method robustness.
• The approach facilitates constructing high-order finite elements on immersed boundaries without requiring mesh conformity, thereby automating mesh generation for complex geometries.

Vertex-sharing mesh parameterization refers to a class of methods for mapping planar curves, particularly $C^2$-regular boundaries $\Gamma$, onto structured representations within triangulated ambient spaces. The approach enables the parameterization of $\Gamma$ via closest point projection over a discrete set of mesh edges $\Gamma_h$, utilizing only a straight-edge triangulation of a polygonal superset of the domain. Rigorous geometric criteria guarantee that the resulting piecewise parameterization is a global homeomorphism that is $C^1$ on each edge, with robust control over the Jacobian. This provides a foundation for constructing curved finite elements and high-order methods on immersed boundaries without mesh conformity requirements (Rangarajan et al., 2011).

1. Construction of the Discrete Edge Set $\Gamma_h$

Let $\Gamma \subset \mathbb{R}^2$ be a $C^2$-regular closed curve and $\mathcal{T}_h$ a (possibly nonconforming) straight-edge triangulation containing $\Gamma$. The signed distance function $\Gamma$0 is used to classify mesh vertices: $\Gamma$1 for vertices inside $\Gamma$2, $\Gamma$3 for vertices outside. A triangle $\Gamma$4 is positively-cut if exactly two vertices satisfy $\Gamma$5 and the third $\Gamma$6. Edges $\Gamma$7 with both endpoints outside ($\Gamma$8, $\Gamma$9) and contained in a positively-cut triangle are positive edges. The union of all such positive edges forms $\Gamma$0: $\Gamma$1 This definition ensures that $\Gamma$2 consists of mesh edges “sharing” vertices on a single side of $\Gamma$3 while lying adjacent to triangles “cut” by the curve.

2. Geometric and Mesh Quality Criteria

Accurate parameterization and injectivity require localized geometric conditions around $\Gamma$4. For each positively-cut triangle:

• $\Gamma$5 is the diameter, $\Gamma$6 the in-radius, and $\Gamma$7 the shape parameter.
• The two “positive” vertices $\Gamma$8 are identified, with $\Gamma$9 termed the proximal vertex if $\Gamma_h$0.
• $\Gamma_h$1 angle at $\Gamma_h$2 in $\Gamma_h$3 (conditioning angle); $\Gamma_h$4 is the minimum of the two adjacent angles from the neighboring triangle sharing $\Gamma_h$5.
• $\Gamma_h$6 is the supremum of curvature in the local region.
• $\Gamma_h$7 provides a condition number for the local curvature and triangle size.

Sufficient conditions for each $\Gamma_h$8 are:

1. $\Gamma_h$9 (with $C^1$0 the regularity radius for $C^1$1 and $C^1$2),
2. $C^1$3,
3. $C^1$4,
4. $C^1$5.

These criteria ensure robustness of the closest point projection and the exclusivity of positive edges per triangle.

3. Theoretical Properties: Homeomorphism and Regularity

The main theorem of Rangarajan & Lew (Rangarajan et al., 2011) guarantees that, under the above mesh and geometric restrictions:

• Every positive edge in $C^1$6 belongs to exactly one positively-cut triangle.
• Each positive edge $C^1$7, restricted to its relative interior, is mapped by $C^1$8 as a $C^1$9-diffeomorphism.
• The signed distance along $\Gamma_h$0 is bounded:

$\Gamma_h$1

• The Jacobian $\Gamma_h$2 satisfies

$\Gamma_h$3

where $\Gamma_h$4 and $\Gamma_h$5.

• Globally, $\Gamma_h$6 is a homeomorphism and $\Gamma_h$7 is a disjoint union of Jordan loops, each covering a connected component of $\Gamma_h$8.

Propositions guarantee that the edgewise $\Gamma_h$9 regularity and injectivity transfer to global properties via the loop structure of $\Gamma \subset \mathbb{R}^2$0.

4. Algorithmic Implementation and Practical Considerations

A prototypical implementation proceeds through the following steps: A. Preprocessing: Compute the signed-distance $\Gamma \subset \mathbb{R}^2$1 and its gradient at each vertex; evaluate the signs on all triangles. B. Positive Edge Detection: Mark positively-cut triangles; identify positive edges by their endpoints and adjacent triangles. C. Mesh Quality Checks: For each positively-cut triangle, evaluate $\Gamma \subset \mathbb{R}^2$2, $\Gamma \subset \mathbb{R}^2$3, $\Gamma \subset \mathbb{R}^2$4, proximal vertex, angles, estimate $\Gamma \subset \mathbb{R}^2$5 (via local curvature sampling or analytic bound), compute $\Gamma \subset \mathbb{R}^2$6, and check all geometric conditions. D. Assembly and Parameterization: Construct the (possibly non-simple) graph of positive edges, extract connected components as loops (Jordan curves), and for each edge $\Gamma \subset \mathbb{R}^2$7, parameterize $\Gamma \subset \mathbb{R}^2$8, $\Gamma \subset \mathbb{R}^2$9 with $C^2$0. E. Jacobian Diagnostics (Optional): Compute $C^2$1, $C^2$2, confirm alignment with the theoretical bounds on a sampling of each edge.

This workflow is explicitly parallel over triangles/edges, requires only standard mesh data structures, and is independent of mesh conformity to the curve.

5. Mathematical Tools: Closest-Point Projection and Regularity

For a $C^2$3-regular curve $C^2$4 with associated tubular neighborhood $C^2$5, both signed-distance $C^2$6 and closest point projection $C^2$7 are well-defined and $C^2$8 within $C^2$9. Specifically, $\mathcal{T}_h$0 satisfies $\mathcal{T}_h$1, and

$\mathcal{T}_h$2

where $\mathcal{T}_h$3 is the unit tangent and $\mathcal{T}_h$4 the signed curvature at $\mathcal{T}_h$5. On every positive edge $\mathcal{T}_h$6, $\mathcal{T}_h$7 ensures $\mathcal{T}_h$8 regularity. Jacobian lower bounds are maintained via the acute angle restrictions and upper bounds by curvature locally, thereby supporting stable mapping for high-order finite element methods.

6. Applications and Significance in Finite Element Methods

This vertex-sharing parameterization method underpins the construction of high-order curved finite elements for domains defined by $\mathcal{T}_h$9 (including piecewise $\Gamma$0) planar curves, as demonstrated in (Rangarajan et al., 2011). Its key features are that:

• No mesh vertices are required to lie on $\Gamma$1.
• $\Gamma$2 can be parameterized (by closest point projection) using only the vertices and edges from a background triangulation.
• The global homeomorphism and $\Gamma$3 regularity ensure accurate trace approximation and quadrature for finite element assembly.

A plausible implication is enhanced automation in mesh generation for complex geometries, as the method’s criteria are compatible with standard meshers and can be checked algorithmically without special meshing around $\Gamma$4.

7. Summary Table of Edge Classification

Criterion	Condition	Mesh Element
Positive edge	Both vertices $\Gamma$5, adjacent to negatively-signed vertex	Mesh edge
Positively-cut triangle	Two vertices $\Gamma$6, one $\Gamma$7	Mesh triangle
Acute conditioning angle	$\Gamma$8	Triangle angle
Size and curvature conditioning	$\Gamma$9, $\Gamma$00	Triangle scalar

The vertex-sharing mesh parameterization is thus a rigorously-defined, computationally efficient approach for geometry-immersed parameterizations with robust theoretical guarantees and direct application to high-order numerical methods (Rangarajan et al., 2011).