Set operations on polyhedra using binary space partitioning trees
SIGGRAPH '87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques
Marching cubes: A high resolution 3D surface construction algorithm
SIGGRAPH '87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques
Polygonization of implicit surfaces
Computer Aided Geometric Design
The algorithmic beauty of plants
The algorithmic beauty of plants
Octrees for faster isosurface generation
ACM Transactions on Graphics (TOG)
Multiresolution analysis for surfaces of arbitrary topological type
ACM Transactions on Graphics (TOG)
Complex-valued contour meshing
Proceedings of the 7th conference on Visualization '96
Isosurfacing in span space with utmost efficiency (ISSUE)
Proceedings of the 7th conference on Visualization '96
Interval volume tetrahedrization
VIS '97 Proceedings of the 8th conference on Visualization '97
Computing the separating surface for segmented data
VIS '97 Proceedings of the 8th conference on Visualization '97
Isosurfacing in higher dimensions
Proceedings of the conference on Visualization '00
Proceedings of the conference on Visualization '01
Geometric Substitution: A Tutorial
IEEE Computer Graphics and Applications
Sweeping Simplices: A Fast Iso-Surface Extraction Algorithm for Unstructured Grids
VIS '95 Proceedings of the 6th conference on Visualization '95
Span filtering: an optimization scheme for volume visualization of large finite element models
VIS '91 Proceedings of the 2nd conference on Visualization '91
Pixel-Exact Rendering of Spacetime Finite Element Solutions
VIS '04 Proceedings of the conference on Visualization '04
Volume Interval Segmentation and Rendering
VV '04 Proceedings of the 2004 IEEE Symposium on Volume Visualization and Graphics
IEEE Transactions on Visualization and Computer Graphics
Visualization in Medicine: Theory, Algorithms, and Applications
Visualization in Medicine: Theory, Algorithms, and Applications
Preserving zeros in surface construction using marching cubes
Machine Graphics & Vision International Journal
VG'05 Proceedings of the Fourth Eurographics / IEEE VGTC conference on Volume Graphics
Discrete multi-material interface reconstruction for volume fraction data
EuroVis'08 Proceedings of the 10th Joint Eurographics / IEEE - VGTC conference on Visualization
Visualization and analysis-oriented reconstruction of material interfaces
EuroVis'10 Proceedings of the 12th Eurographics / IEEE - VGTC conference on Visualization
Interactive visualization of function fields by range-space segmentation
EuroVis'09 Proceedings of the 11th Eurographics / IEEE - VGTC conference on Visualization
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We describe how to count the cases that arise in a family of visualization techniques, including Marching Cubes, Sweeping Simplices, Contour Meshing, Interval Volumes, and Separating Surfaces. Counting the cases is the first step toward developing a generic visualization algorithm to produce substitopes (geometric substitutions of polytopes). We demonstrate the method using a software system ("GAP") for computational group theory. The case-counts are organized into a table that provides a taxonomy of members of the family; numbers in the table are derived from actual lists of cases, which are computed by our methods. The calculation confirms previously reported case-counts for large dimensions that are too large to check by hand, and predicts the number of cases that will arise in algorithms that have not yet been invented.