Related work: Contributing Vertices-Based Minkowski Sum of a
Non-Convex–Convex Pair of Polyhedra
Authors: Hichem Barki, Florence Denis, and Florent Dupont
Note
All files arre provided in off file format. We recommend JavaView to see the models.
We also recommend the Mozilla Firefox web browser to see our webpage.
We provide the following sets of results:
Sample results 1: Minkowski sum of several non-convex--convex pairs of polyhedra computed by the NCC-CVMS algorithm.
- You can click on the different images to zoom in.
- If you want to download a model containing the input polyhedra and the result of the NCC-CVMS algorithm, you have just to click on the Download link inside each cell.
- Each downloaded .zip file is named NCC-CVMS-result.zip. This .zip file contains the following .off files:
- A.off contains the non-convex polyhedron A (e.g. Aircraft.off, Bunny.off, Grate2.off, etc.)
- B.off contains the convex polyhedron B (e.g. Cube_quad.off, Sphere.off, TruncatedOctahedron.off, etc.)
- MS_superset.off contains the reduced MS superset generated from the input polyhedra
- MS_superset_merged_triangulated.off contains the MS superset after the merging and the triangulation operations
- MS_seed_facet.off contains the seed facet (the first MS facet computed by the NCC-CVMS algorithm)
- MS_polyhedron.off contains the MS polyhedron computed by the NCC-CVMS algorithm
- If you are impatient to see some of our models without downloading them, try some of the following links that allow you to visualize the models directly in your web browser (need Java enabled on your machine).
Sample results 2: This .zip file contains examples of Minkowski sums exhibiting non-manifold situations and polyhedral holes computed succesfully by our NCC-CVMS algorithm.
Sample results 3: This NCC-CVMS-vs-Nef.zip file includes two folders: the first one contains some MS polyhedra computed by our NCC-CVMS algorithm and the second contains the same MS polyhedra computed by the Nef polyhedra-based approach. You can download the Metro software to compute the distance between the MS polyhedra computed by the two approaches.
Sample results 4: We provide some movies showing the traversal of the MS boundary as performed by the NCC-CVMS algorithm.
Some of the reported models are taken from http://masc.cs.gmu.edu/wiki/SimpleMsum
Sample results 1: Minkowski sum of several non-convex / convex pairs of polyhedra
|
|
B
|
 Cube_quad |
 SnubCube |
 Sphere |
 TruncatedTetra. |
|---|---|---|---|---|---|
|
A
|
A+B
|
||||
 Aircraft |
 A+B |
 A+B |
 A+B |
 A+B |
|
|
|
B
|
 Cube_quad |
 Sphere |
 TruncatedDodeca. |
 TruncatedTetra. |
|---|---|---|---|---|---|
|
A
|
A+B
|
||||
 Armchair |
 A+B |
 A+B |
 A+B |
 A+B |
|
|
|
B
|
 Cube_quad |
 Icosahedron |
 RhombicTriaconta. |
 Sphere |
|---|---|---|---|---|---|
|
A
|
A+B
|
||||
 Axes |
 A+B |
 A+B |
 A+B |
 A+B |
|
|
|
B
|
 Cube_quad |
 IcosiDodeca. |
 RhomIcosiDodeca. |
 TruncatedOcta. |
|---|---|---|---|---|---|
|
A
|
A+B
|
||||
 Baby |
 A+B |
 A+B |
 A+B |
 A+B |
|
|
|
B
|
 Cube_quad |
 SnubCube |
 Sphere |
 TruncatedIcosa. |
|---|---|---|---|---|---|
|
A
|
A+B
|
||||
 Bunny |
 A+B |
 A+B |
 A+B |
 A+B |
|
|
|
B
|
 Cube_quad |
 Rh.Dodeca. |
 Sphere |
 Trunc.Icosa. |
|---|---|---|---|---|---|
|
A
|
A+B
|
||||
 Chair |
 A+B |
 A+B |
 A+B |
 A+B |
|
|
|
B
|
 Cube_quad |
 Icosahedron |
 SnubDodeca. |
 Sphere |
|---|---|---|---|---|---|
|
A
|
A+B
|
||||
 Dinausor |
 A+B |
 A+B |
 A+B |
 A+B |
|
|
|
B
|
 Cube_quad |
 RhombicDodeca. |
 Sphere |
 TruncatedTetra. |
|---|---|---|---|---|---|
|
A
|
A+B
|
||||
 Grate1 |
 A+B |
 A+B |
 A+B |
 A+B |
|
|
|
B
|
 RhombicDodeca. |
 SnubCube |
 Sphere |
 TruncatedTetra. |
|---|---|---|---|---|---|
|
A
|
A+B
|
||||
 Grate2 |
 A+B |
 A+B |
 A+B |
 A+B |
|
|
|
B
|
 Cube_quad |
 RhombiCubeOcta. |
 Sphere |
 TruncatedTetra. |
|---|---|---|---|---|---|
|
A
|
A+B
|
||||
 Hand_man |
 A+B |
 A+B |
 A+B |
 A+B |
|
|
|
B
|
 Cube_quad |
 RhombiTr.Cube |
 Sphere |
 TruncatedTetra. |
|---|---|---|---|---|---|
|
A
|
A+B
|
||||
 Horse |
 A+B |
 A+B |
 A+B |
 A+B |
|
|
|
B
|
 Cube_quad |
 Icosahedron |
 RhombicTriaconta. |
 Sphere. |
|---|---|---|---|---|---|
|
A
|
A+B
|
||||
 Knot |
 A+B |
 A+B |
 A+B |
 A+B |
|
|
|
B
|
 Cube_quad |
 RhombiTr.Cube |
 Sphere |
 TruncatedTetra. |
|---|---|---|---|---|---|
|
A
|
A+B
|
||||
 L |
 A+B |
 A+B |
 A+B |
 A+B |
|
|
|
B
|
 Cube_quad |
 RhombiTr.Icosa. |
 Sphere |
 TruncatedTetra. |
|---|---|---|---|---|---|
|
A
|
A+B
|
||||
 Path |
 A+B |
 A+B |
 A+B |
 A+B |
|
|
|
B
|
 Cube_quad |
 Sphere |
 TruncatedTetra. |
 TruncatedCube |
|---|---|---|---|---|---|
|
A
|
A+B
|
||||
 Pig |
 A+B |
 A+B |
 A+B |
 A+B |
|
|
|
B
|
 Cube_quad |
 Rh.Ic.Dodeca. |
 Sphere |
 Trunc.Dodeca. |
|---|---|---|---|---|---|
|
A
|
A+B
|
||||
 Rabbit |
 A+B |
 A+B |
 A+B |
 A+B |
|
|
|
B
|
 Cube_quad |
 RhombiTr.Cube |
 Sphere |
 TruncatedTetra. |
|---|---|---|---|---|---|
|
A
|
A+B
|
||||
 Skull |
 A+B |
 A+B |
 A+B |
 A+B |
|
|
|
B
|
 Cube_quad |
 Sphere |
 TruncatedOcta. |
 TruncatedTetra. |
|---|---|---|---|---|---|
|
A
|
A+B
|
||||
 Star |
 A+B |
 A+B |
 A+B |
 A+B |
|
|
|
B
|
 Cube_quad |
 RhombiTr.Icosa. |
 Sphere |
 TruncatedTetra. |
|---|---|---|---|---|---|
|
A
|
A+B
|
||||
 Wrench |
 A+B |
 A+B |
 A+B |
 A+B |
|
Sample results 4: Some movies showing how the NCC-CVMS algorithm traverses the MS boundary
The NCC-CVMS boundary traversal of the Minkowski sum of the Aircraft model and the Cube model
The NCC-CVMS boundary traversal of the Minkowski sum of the Grate1 model and the TruncatedTetrahedron model
The NCC-CVMS boundary traversal of the Minkowski sum of the Star model and the Sphere model