EuroCG 2011, Morschach, Switzerland, March 28–30, 2011

High Quality Surface Mesh Generation for Swept Volumes∗

Andreas von Dziegielewski † Michael Hemmer‡

Abstract

We present a novel and efficient technique to gener-ate a high quality mesh that approximates the outerboundary of a swept volume (SV). Our approachcomes with two guarantees. First, the approxima-tion is conservative, i.e. the swept volume is enclosedby the output mesh. Second, the one-sided Hausdorffdistance of the generated mesh to the swept volume isupper bounded by a user defined tolerance. Exploit-ing this tolerance our method produces an anisotropicmesh which nicely adapts to the local complexity ofthe approximated swept volume boundary. The algo-rithm is two phased: a initialization phase that gen-erates a conservative voxelization of the swept vol-ume, and the actual mesh generation which is basedon CGAL’s Delaunay refinement implementation.

1 Introduction

The swept volume is defined as the entity of all pointstouched by a solid (the generator) under the transfor-mations of either a continuous or discrete trajectory.In the context of this paper, we assume a discretetrajectory and define SV as the entity obtained bylinear interpolating the generator geometry betweenconsecutive time steps, as proposed in [2].

The swept volume plays an important role in com-puter aided design (CAD), numerically controlled(NC) machining verification, robot analysis andgraphical modeling. For safety reasons most applica-tions demand for a conservative approximation of SV,that is, the result must contain SV. At the same time,the result should approximate SV as close as possi-ble while keeping the complexity of the output low,the latter becoming more and more relevant due toincreasing model complexity and high demands con-cerning error tolerance. Moreover, a practical algo-rithm should be tolerant to topologically inconsistentinput data and should preferably impose no restric-tions on the input models.

∗This work has been supported in part by the 7th Frame-work Programme for Research of the European Commission,under FET-Open grant number 255827, CGL, ComputationalGoemetry Learning.†Institut fur Informatik, Johannes Gutenberg Universitat

Mainz, dziegiel@uni-mainz.de‡School for Computer Science, Tel Aviv University,

mhsaar@google.mail.com

1.1 Previous and related work

Mathematical formulations describing the swept vol-ume include Jacobian rank deficiency methods, sweepdifferential equations and envelope theory, for a sur-vey see [1].Practical approaches, most relying on volumetric SVrepresentations cannot give geometrical guaranteesbecause sharp features can be missed ([7, 6]). In [11]the authors claim to be able to archieve arbitrarilytight error-bounds (although do not regard conser-vativeness) but unfortunately no timings for a priorierror bounds are given. They all produce a highlyovertessallated mesh and do not propose any errorbound mesh simplification method. Applying errorbound simplification (e.g. [10]) to these meshes, a pos-teriori, will always be limited by the storage neededfor the mass of triangles of the original mesh.

Recent approaches [9, 3] are able to produce a con-servative approximation. In [3] local culling criteriaare applied to generate a superset of the SV bound-ary that is then blended using a BSP tree. Their out-put mesh is non-adaptive and possibly highly overtes-sellated, and further simplification in convex regionswould violate conservativeness. The method pro-posed in [9] relies on special conservative depth buffervoxelization and an error bound mesh simplificationphase. The output mesh is adaptive and almost every-where manifold. It is conservative, but error boundscan not be given for all concave parts of the SV.

Figure 1: Bunny swept along a trajectory consistingof 12 transformations.

27th European Workshop on Computational Geometry, 2011

1.2 Our contribution

In contrast to these previous approaches our mesh isbuild top down, i.e. we start from a coarse mesh andapply local refinement until we are guaranteed a con-servative and error bound approximation of the outerSV boundary. More precisely, we guarantee that theone-sided Hausdorff distance to SV does not exceedthe user defined tolerance δ. In addition the result-ing anisotropic mesh is of high quality while keepingthe complexity relatively low, that is: produced trian-gles obey quality constrains such as lower bounds onsmallest angles while mesh density adapts to the localcomplexity of SV as far as it is required by δ. Thealgorithm is two phased: a initialization phase thatgenerates a conservative voxelization of the swept vol-ume, and the actual mesh generation which is basedon CGAL’s Delaunay refinement implementation [8].

The Delaunay refiment and the general scheme ofthe approach are discussed in Section 2. More detailsof the initialization phase, are then discussed in Sec-tion 3. Implementation details and some preliminaryresults are presented in Section 4.

2 Overview

In order to generate the output mesh we apply De-launay refinement which is known to be one of themost powerful techniques in the field of mesh gener-ation and surface approximation. More precisely, ouralgorithm utilizes CGAL’s frame work for 3D meshgeneration [8] which we briefly review next.

Starting from an initial points set on the surface ofthe to be meshed domain D, the process maintains aDelaunay triangulation of this point set. Thereby, itclassifies each tetrahedron as interior or exterior ac-cording to the position of the center of its circumscrib-ing ball. A triangle is said to belong to the surfaceif the classification of the two neighboring tetrahedradiffer. Such a surface facet f can be refined by induc-ing the intersection point of its Voronoi edge (the dualof f) with ∂D.1 The refinement process now succes-sively refines those boundary facets that are classifiedas bad facets, for instance, triangles that are consid-ered two large or whose minimal angle is too small.The latter criterion ensures well formed triangles inthe resulting mesh.

The main idea of our approach is to add anothercriterion that classifies a boundary facet as bad if: (a)the facet intersects SV or (b) the one-sided Hausdorff-distance of the facet to SV is too large. Note thatpart (a) implies that we can not place intersectionpoints directly on ∂SV since boundary facets wouldalways intersect SV in regions where SV is locallyconvex. Thus the idea is to place intersections points

1On the duality of the Delaunay triangulation and theVoronoi diagram see for instance [4]

Figure 2: Sweeping a triangle yields a prism withthree ruled surfaces, each of which is approximatedand tessellated by inserting the diagonal with thelower dihedral angle.

on some offset of SV at about half of the user definedtolerance. This way it is guaranteed that boundaryfacets eventually fulfill both conditions which in turnguarantees termination of the generation process.

The approach is two phased: In a initializationphase we generate a conservative and sufficiently pre-cise voxelization V0 of SV. In addition we computetwo offsets V1 and V2 by successively adding an ad-ditional layer of voxels to V0. In the second phase,the mesh generation phase, we set D = V1. Intersec-tion points of Voronoi edges with ∂V1 are computedusing bisection. In order to classify a facet, the facetis voxelized. The facet is considered as bad if one ofits voxels is contained in V0 (ensuring (a)) or outsideof V2 (ensuring (b)).

Thus the core of our approach is an efficient gener-ation of V0 and its offsets, which is discussed next.

3 Initialization Phase

In order to obey the user defined tolerance δ it isof course necessary to chose the size of each voxel εsmaller than δ. Taking into account that the diagonalof a voxel is ε

√3, the two layers for V2 as well as an-

other layer for the conservative voxelization of V0 wehave to chose ε < δ/3

√3. That is, for a reasonable

scenario with an area of interest of about 1m3 anda user defined tolerance of 1mm this would result inabout 52003 ' 140 ∗ 109 voxel, which brings memoryconsumption onto the table. In order to decrease thememory usage we store a voxelization as an octreewhich we, similar to [5], internally represented by ahash set. A node is marked as occupied by its pureexistence in the hash set. In case all 8 children of anode are marked the node is marked and the childrencan be deleted. A voxel is not covered if its leaf andnon of its ancestors exists in the hash set. Testing con-tainment as well as insertion is in O(log 1/ε) However,the most important property is that for a reasonableSV one can expect a memory consumption that isproportional to ∂SV, i.e., quadratic in 1/ε.

Assuming a linear interpolation every triangle givesrise to a deformed prism between two trajectory po-sitions, see also Figure 2. Thus inserting all voxels of

EuroCG 2011, Morschach, Switzerland, March 28–30, 2011

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Figure 3: Total time in relation to main functions.

a conservative voxelization of each such prism woulddirectly yield V0. Since generating all these voxels isto expensive we follow a culling heuristic by [3] thattries to rule out those triangles (of the prisms) that donot contribute to the boundary. This is based on localcriteria of the mesh in relation to the current directionof the motion. Only the remaining triangles are thenvoxelized. The voxelization is based on a simple sub-division scheme. Starting from the initial cube, theboxes that still intersect the triangle are subdivideduntil the required resolution is reached. The intersec-tion test is based on the separating axes theorem. Theresult ∂V0 is a subset of V0 and contains ∂V0. In anext step we generate ∂V1 = V1\V0 by crawling alongthe outer boundary of ∂V0. We then throw away ∂V0and generate V0 by filling ∂V1. V1 is then generatedby simply adding ∂V1 to a copy of V0. V2 is created ina similar fashion. Since ∂V0 and ∂V1 are stored in aplain hash set the crawling is, for reasonable volumes,in O((1/ε)2). Note that the filling is not cubic sinceit uses an hierarchic scheme, which we can no discusshere due to limited space.

4 Preliminary Results

All benchmarks were measured on a Intel(R)Core(TM) i5 CPU M 450 with 2.40GHz with 512 kBcache under Ubuntu Linux and the GNU C++ com-piler v4.4 with optimizations (-O2). However, we onlyuse one CPU since we did not parallelize any part ofthe approach yet.

The swept Stanford Bunny, Figures 1 and 5, weregenerated from an initial mesh with 8100 triangles andtrajectories of size 12/50. The resolution was set to210 = 1024 which corresponds to δ ' 0.005. One cansee that the mesh nicely adapts to the local complex-ity of SV, with sparse areas in well behaved regions.The resulting mesh has only about 20k/10k bound-ary facets. Figure 3 shows the obtained times for theStanford Bunny shown in Figure 1. Note that times

0.001

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Figure 4: Times for initialization phase.

and resolution are given in a logarithmic scale. Thegraphs show the total time in relation to three ma-jor components: the time for the initialization phase,time spend in point classification and facet classifica-tion. The sum of the later two is essentially the timespend in the mesh generation phase. Thus already forsuch a small example the initialization phase clearlydominates the time used overall.

It turns out that most of the time of the initializa-tion phase is currently spend in the voxelization of alltriangles that where not culled and the computationof the hull surrounding it. All other steps such as thehierarchical filling of V0 and the offset computationto compute V1 and V2 from V0 are not relevant. Fig-ure 4, shows a more detailed plot of the time spendin the hull computation. Initially, the voxelization isclearly the dominating part, but for high resolutionthe hull computation eventually takes over. This isdue to the fact that the offset computation in thispart is more expensive since the volume is not filledand thus the size of the octree is much larger. How-ever, for larger models and larger trajectories the firstpart would currently be the most dominating timefactor.

5 Further Work

Though our results are already very promising wealready see several, partially obvious, optimizationsthat where not yet applied due to lack of time.

Using only one offset would significantly improvememory usage but would also improve ε to δ/4. Itonly requires a slightly modified point generationfunction. On the other hand, the new scheme wouldonly leave a corridor of width about δ/2 to place tri-angles. This would probably have a negative impacton the size of the resulting mesh. For the currentscheme the width is about 2δ/3.

It is clear that the voxel generation can be easilyparallelized. For instance, each CPU may be ded-

27th European Workshop on Computational Geometry, 2011

icated to a certain volume in space. The resultingoctrees may be merged afterwards. That is, it is easyto parallelize the currently most time consuming partof the algorithm. An obvious alternative is to movethe voxel generation to the GPU.

So far our implementation can not handle inputmeshes that are not manifolds. However, this is justdue to missing case distinction in the current codehandling the culling. In principal the approach is ableto easly handle meshes that are not watertight or haveother topological inconsistencies. For instance, thevoxelization would fill the gap if it is smaller than ε.Note that this also means that the approach ignoresnarrow tunnels to maybe rather large caves inside themodel. In case the tunnel has diameter around deltathe hull computation explores the cave, but it mayappear as a closed void since the tunnel is closed dueto the subsequent offset computations. Note that thisdoes not contradict our guarantees since we are onlyinterested in the one-sided Hausdorff-distance.

Figure 5: Bunny swept along a trajectory consistingof 50 transformations.

6 Conclusion

To the best of our knowledge this is the first con-servative approache for swept volume mesh geneara-tion that also guarantees an approximation quality interms of the one-sided Hausdorff distance. The result-ings meshes are of high quality and very reasonablesize, which makes them ideal for further processingin industrial applications. Moreover, our prelimiarybenchmark results indicate that the approach shouldeven applicalbe for very large inputs, in particular,once the parallization of the initialization phase is inplace.

For recent improvements, more examples andother supplementary material we refer to our

website: http://acg.cs.tau.ac.il/projects/

internal-projects/swept-volume/.

Acknowledgments

We thank R. Erbes for supplying us with the voxelization

algorithm for triangles.

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