Open research questions in Computational Geometry and Mesh Generation
85 unresolved questions extracted from the limitations and future-work sections of 333 Computational Geometry and Mesh Generation papers in our library. Each links back to the study that raised it.
What the literature leaves open
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 gap in prior work is that it was not known whether an optimal solution of size two might need to be irrational. The gap is addressed by showing that an optimal solution of size two might need to be irrational.
To further evaluate the framework's effectiveness. To apply the framework to other applications. To improve the framework's performance and scalability.
Graph-Theoretic and Computational Geometry Approaches for the Management of Sensor Networks in Process Monitoring · 2026 · DOIThe lack of a generalization of splitting invariants for plane curves. The need for a new method to distinguish the embedded topology of plane curves.
The management of sensor networks is a complex task. Prior work has not fully addressed this task.
Graph-Theoretic and Computational Geometry Approaches for the Management of Sensor Networks in Process Monitoring · 2026 · DOIFurther study of the properties of the 𝐺-combinatorial type. Application of the 𝐺-combinatorial type to other areas of research.
The paper identifies challenges related to the application of integral calculus in various fields. It highlights the importance of understanding integral calculus in solving real-world problems. The challenges are limited to the theoretical context presented in the paper.
The paper is limited to a theoretical analysis of integral calculus and its applications. It does not provide specific numerical results or data to support the presented concepts. The paper acknowledges the importance of understanding integral calculus but does not address potential limitations or challenges in its application.
Applying the algorithm to other types of data, such as economic or social variables. Exploring the use of different force field concepts or distance decay functions. Investigating the application of the algorithm to different geographic regions or scales.
Previous algorithms had limitations in handling realistic descriptions of polygon boundaries. Previous algorithms had slow convergence and limited accuracy.
The problem of finding the minimum number of facilities needed to cover a set of demand points. The challenge of solving the problem with facility placement on the entire plane.
The lack of a method to solve the maximal covering location problem with facility placement on the entire plane. The need for a dynamic programming approach to solve the problem.
The lack of a systematic approach to constructing triangles with located points is a gap in prior work. The paper identifies the need for a method that can reconstruct the original figure with certain triples of located points.
The lack of guidance about how the review process should be tackled. The need to balance the level of LEA involvement in the exercise. The potential for the heavy LEA involvement to invalidate the recommendations made in the materials.
The possibility of adopting a more overt training function for the LEA coordinators. The need to consider the validity of the recommendations made in the materials. The potential for the GRIDS materials to be used in other contexts.
The need for algorithms that can approximate a connected set of fixed points. The lack of a triangulation of R^n that allows for quick movements in all directions.
Future research could explore the use of Gabriel graphs in other fields, such as computer science or biology. Future research could develop new methods for clustering points in the plane using Gabriel graphs.
Properties of Gabriel Graphs Relevant to Geographic Variation Research and the Clustering of Points in the Plane · 1980 · DOIThe paper identifies a gap in the literature regarding the planarity of Gabriel graphs. The paper identifies a need for a new characterization of Gabriel graphs using the least squares adjacency criterion.
Properties of Gabriel Graphs Relevant to Geographic Variation Research and the Clustering of Points in the Plane · 1980 · DOIThe use of proximity-based triangulations as storage structure for geographic information systems. The use of the procedure for complex mapping tasks.
The paper identifies a gap in the understanding of the relation between cognitive development and object relations in children - The paper highlights the need for further research on the use of play therapy in understanding children's anxieties
The effect of the boundary on the analysis is a major problem. There is a need to evaluate procedures to obtain the distribution of distances to the jth nearest neighbour. There is a need to compare the sensitivity of alternative tests of the null hypothesis.
Extend the results to more general types of graphs. Develop formulas for counting the numbers of tours and paths possible for an h x k RLG.
The gap involves finding better arrangements for larger values of n. The derivation of lower bounds for the area of the smallest triangle is limited to specific cases.
Further analysis of the types of lattice points and edges may provide insights into the problem. The study of geometric dissections may lead to new results.
The method is limited to solving the specific problem of crossing a river with cannibals and missionaries. The paper does not provide a general solution to all "Difficult Crossing" puzzles. The method involves tedious but direct computation of the matrix product.
Most-cited papers in Computational Geometry and Mesh Generation
- The new demarcation problem · Studies in History and Philosophy of Science Part A · 2021 · 102 citations
- Weighting Thiessen Polygons · Economic Geography · 1980 · 73 citations
- An interactive user interface for a structural analysis software using computer graphics techniques in MATLAB · Computer Applications in Engineering Education · 2021 · 20 citations
- Additional considerations and response to ‘graphical representation of overlap for OVErviews (GROOVE tool)’ · Research Synthesis Methods · 2022 · 8 citations
- No one solution to the “new demarcation problem”?: A view from the trenches · Studies in History and Philosophy of Science Part A · 2022 · 6 citations
- Geometric prediction: A framework to gain insight into solvers’ geometrical reasoning · The Journal of Mathematical Behavior · 2021 · 6 citations
- Can a number of diagrams linked to a proof task in 3D geometry improve proving ability? · Mathematics Education Research Journal · 2021 · 6 citations
- On face irregular evaluations of plane graphs · Discussiones Mathematicae Graph Theory · 2020 · 4 citations
- On The Voronoi Diagram and Taxicab Plane · Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi · 2021 · 3 citations
- A Differentiable Mapping of Mesh Cells Based on Finite Elements on Quadrilateral and Hexahedral Meshes · Computational Methods in Applied Mathematics · 2020 · 3 citations
Most recent work
- Sometimes Two Irrational Guards are Needed · Discrete Mathematics & Theoretical Computer Science · 2026
- A Novel Approach for Quad‐Dominant Mesh Generation Using Conformal Mapping Compressed Gradient Field · International Journal for Numerical Methods in Engineering · 2026
- Surface Area Minimization in Pyramids over Tangential Polygons · Mathematics Magazine · 2026
- Graph-Theoretic and Computational Geometry Approaches for the Management of Sensor Networks in Process Monitoring · Afinidad. Journal of Chemical Engineering Theoretical and Applied Chemistry · 2026
- A note on combinatorial type and splitting invariants of plane curves · Canadian Journal of Mathematics · 2026
- ON STRUCTURAL AND ALGEBRAIC PROPERTIES OF EDGE GEODETIC CLOSURE POLYNOMIALS IN GRAPHS · Advances and Applications in Discrete Mathematics · 2026
- Approximation Algorithm for Minimum p Union under a Geometric Setting · Journal of the Operations Research Society of China · 2026
- North–East lattice paths avoiding k collinear points via satisfiability · Advances in Applied Mathematics · 2026
- Constrained Shortest Path Finding on Terrain Surfaces · Proceedings of the ACM on Management of Data · 2026
- Tight Runtime Bounds for Evolutionary Algorithms on Sorting and Crossing Minimisation for Layered Graph Drawings · Algorithmica · 2026
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