Open research questions in Computational Geometry and Mesh Generation
101 gap statements mined from Computational Geometry and Mesh Generation papers in our 4.5M-paper local library, which holds 580 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.
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What the literature leaves open
Dealing with the lack of regularity of the sets involved. Developing new techniques to improve bounds on the Hausdorff dimension of pinned distance sets. Extending the results to more general classes of sets.
The lack of a sharp upper bound for an antipodal partial covering of the sphere. The limitations of existing techniques for solving the problem. The need for a new approach to the problem.
Prior work has focused on Euclidean spaces, leaving a gap in the development of spherical bounding shapes. Global-scale applications require bounding shapes that preserve geodesic distances and avoid projection distortion. Existing methods introduce distortions, particularly near the poles or along coordinate axes.
Fractal theory provides a mathematical tool for characterizing the self-similarity of complex geometric structures; however, a single fractal dimension is insufficient to capture the multiscale heterogeneity of meshes.
Future research can focus on improving the approximation factor of a grid. Future research can focus on finding more efficient algorithms for guard placement.
The lack of a generalization of splitting invariants for plane curves. The need for a new method to distinguish the embedded topology of plane curves.
Future research can focus on improving the approximation ratio of the algorithm. Future research can focus on applying the algorithm to other problems. Future research can focus on providing a polynomial-time algorithm for the minimum p union problem.
The paper identifies a need for a dynamic programming approach to the problem. The paper identifies a need for an improved approximation algorithm.
This solves a long-standing open problem by Scharnow et al. on whether the upper (n2 log n) for sorting with evolutionary algorithms using exchanges or bound of jumps is asymptotically tight. It also shows that, surprisingly, the much simpler swap operator is faster by a factor of Θ(log n) as it sorts any sequence of numbers (n2) with high probability and in and solves every planar OBCM instance in time expectation.
Tight Runtime Bounds for Evolutionary Algorithms on Sorting and Crossing Minimisation for Layered Graph Drawings · 2026 · DOIThe problem of whether the T(3) property implies a finite line-piercing number was unknown. The previous result by McGinnis and Zerbib required 6 families instead of 5.
The lack of a solution for partitioning triangle-free planar graphs into a forest and a linear forest. The need for a coloring algorithm that can achieve this partitioning. The gap in current research on the maximum degree of the forest.
The challenge is to provide a guarantee on the maximum error. The challenge is to develop a computationally efficient and scalable method. The challenge is to address the gap in prior methods.
The lack of a quantitative understanding of the topology of Vietoris-Rips complexes. The need for a technique to reconstruct Euclidean graphs from noisy samples.
The study addresses the lack of knowledge about the problem's behavior for different values of δ.
From Chinese Postman to Salesman and Beyond II: Inapproximability and Parameterized Complexity · 2026 · DOIThe lack of symmetry in non-symmetric planar domains makes it challenging to analyze the torsion function. The complexity of the torsion function and its dependence on the shape and size of the domain.
On the location of the maximal gradient of the torsion function over some non-symmetric planar domains · 2026 · DOIThere is a lack of research on the location of the maximal gradient of the torsion function on non-symmetric planar domains. Prior work has focused on symmetric domains, leaving a gap in the understanding of non-symmetric domains.
On the location of the maximal gradient of the torsion function over some non-symmetric planar domains · 2026 · DOIWithout loss of generality, assume M ≥ N. Within this region, there exists a unique critical line of the form αx + βy = V that passes through the rectangle, where x and y are variables, α, β and V are unknown constants with following constraints. The critical line has a positive y-intercept. The slope of the critical line can be negative, 0, or undefined, so that the line can be descending, horizontal or vertical.
Limited power, memory, and computing resources in embedded systems. Efficient point location and topology maintenance in the SoC-FPGA framework.
Prior methods do not provide guarantees on the maximum error. Prior methods do not address the problem of minimizing the maximum geometric error for a fixed budget of primitives.
The results are limited to planar sets with dimension strictly less than one, - The proofs rely on certain assumptions about the regularity of the planar…
Limited to simulation and implementation on a specific platform (AMD Kria KR260), - No comparison with other hardware implementations, - No discussion of power consumption…
Investigating the sharpness of the bound ln N = o(√d), - Exploring the applications of the Gaussian reduction in other geometric…
To extend the result to more general families of sets. To improve the bound on the number of concurrent lines. To apply the result to other geometric problems.
Further study of the δ-Tour problem's inapproximability and parameterized complexity. Investigation of the problem's behavior for different values of δ. Development of new algorithms for the δ-Tour problem and related problems.
From Chinese Postman to Salesman and Beyond II: Inapproximability and Parameterized Complexity · 2026 · DOIThe paper identifies a gap in the understanding of combinatorial games on graphs, specifically the undirected edge geography game on grid graphs. The authors aim to fill this gap by providing a complete determination of the winning and losing positions for the game on this graph.
Most-cited papers in Computational Geometry and Mesh Generation
- Optimal surface reconstruction from planar contours · Communications of the ACM · 1977 · 579 citations
- Reentrant polygon clipping · Communications of the ACM · 1974 · 454 citations
- Arc and path consistency revisited · Artificial Intelligence · 1986 · 370 citations
- Determining the minimum-area encasing rectangle for an arbitrary closed curve · Communications of the ACM · 1975 · 313 citations
- Algorithms Involving Arithmetic and Geometric Means · American Mathematical Monthly · 1971 · 270 citations
- Properties of Gabriel Graphs Relevant to Geographic Variation Research and the Clustering of Points in the Plane · Geographical Analysis · 1980 · 246 citations
- Nonpartisan Political Redistricting by Computer · Operations Research · 1965 · 234 citations
- Intersection Cuts—A New Type of Cutting Planes for Integer Programming · Operations Research · 1971 · 231 citations
- A Procedure to Generate Thiessen Polygons · Geographical Analysis · 1979 · 224 citations
- Polygons Have Ears · American Mathematical Monthly · 1975 · 151 citations
Most recent work
- Voronoi-based high-quality mesh refinement for tetrahedral mesh generation · Computers & Structures · 2026
- InfinityEBSD : Metrics-guided infinite-size EBSD map generation with diffusion models · Acta Materialia · 2026
- Multi-granularity Granular-ball Anchor Graph Clustering with self-weighting · Information Processing & Management · 2026
- The covering radius of D_d^+ · Zenodo (CERN European Organization for Nuclear Research) · 2026
- Feature-Preserving Offset Meshing · ACM Transactions on Graphics · 2026
- Closed-form solutions for rigid transformations between heterogeneous groups of noisy geometric primitives · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences · 2026
- Multi-pattern structure refinement and fusion for heterophilic graphs · Information Fusion · 2026
- Widths of compact sets in normed spaces and an application in measuring crack widths · Journal of Computational and Applied Mathematics · 2026
- Exact approaches for the pattern minimization problem · Computers & Industrial Engineering · 2026
- MQ-Fus: A novel mesh quality evaluation method based on dual-branch deep fusion network · Computers & Fluids · 2026
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