Properties of n-dimensional triangulations
Computer Aided Geometric Design
Three-dimensional triangulations from local transformations
SIAM Journal on Scientific and Statistical Computing
Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms
ACM Transactions on Graphics (TOG)
Higher-dimensional Voronoi diagrams in linear expected time
Discrete & Computational Geometry
Construction of three-dimensional Delaunay triangulations using local transformations
Computer Aided Geometric Design
Robust adaptive floating-point geometric predicates
Proceedings of the twelfth annual symposium on Computational geometry
Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator
FCRC '96/WACG '96 Selected papers from the Workshop on Applied Computational Geormetry, Towards Geometric Engineering
An empirical comparison of techniques for updating Delaunay triangulations
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Star splaying: an algorithm for repairing delaunay triangulations and convex hulls
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Scalable Parallel Programming with CUDA
Queue - GPU Computing
Parallel geometric algorithms for multi-core computers
Computational Geometry: Theory and Applications
ACMOS'06 Proceedings of the 8th WSEAS international conference on Automatic control, modeling & simulation
Computing 2D constrained Delaunay triangulation using the GPU
I3D '12 Proceedings of the ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games
Flip-flop: convex hull construction via star-shaped polyhedron in 3D
Proceedings of the ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games
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We propose the first algorithm to compute the 3D Delaunay triangulation (DT) on the GPU. Our algorithm uses massively parallel point insertion followed by bilateral flipping, a powerful local operation in computational geometry. Although a flipping algorithm is very amenable to parallel processing and has been employed to construct the 2D DT and the 3D convex hull on the GPU, to our knowledge there is no such successful attempt for constructing the 3D DT. This is because in 3D when many points are inserted in parallel, flipping gets stuck long before reaching the DT, and thus any further correction to obtain the DT is costly. In contrast, we show that by alternating between parallel point insertion and flipping, together with picking an appropriate point insertion order, one can still obtain a triangulation very close to Delaunay. We further propose an adaptive star splaying approach to subsequently transform this result into the 3D DT efficiently. In addition, we introduce several GPU speedup techniques for our implementation, which are also useful for general computational geometry algorithms. On the whole, our hybrid approach, with the GPU accelerating the main work of constructing a near-Delaunay structure and the CPU transforming that into the 3D DT, outperforms all existing sequential CPU algorithms by up to an order of magnitude, in both synthetic and real-world inputs. We also adapt our approach to the 2D DT problem and obtain similar speedup over the best sequential CPU algorithms, and up to 2 times over previous GPU algorithms.