EXACT FACE-OFFSETTING FOR POLYGONAL MESHES

Transcription

1 5.0 GEOMIMESIS/LANDFORMING HAMBLETON + ROSS EXACT FACE-OFFSETTING FOR POLYGONAL MESHES Elissa Ross MESH Consultants Inc. Daniel Hambleton MESH Consultants Inc. ABSTRACT Planar-faced mesh surfaces such as triangular meshes are frequently used in an architectural setting. Face-offsetting operations generate a new mesh whose face planes are parallel and at a fixed distance from the face planes of the original surface. Face-offsetting is desirable to give thickness or layers to architectural elements. Yet, this operation does not generically preserve the combinatorial structure of the offset mesh. Current approaches to this problem are to restrict the geometry of the original mesh to ensure that the combinatorial structure of the underlying mesh is preserved (Pottman and Wallner 2008; Wang and Liu 2010). We present a general algorithm for faceoffsetting polygonal meshes that places no restriction on the original geometry. The algorithm uses graph duality to describe the range of possible combinatorial outcomes at each vertex of the mesh. This approach allows the designer to specify independent offset distances for each face plane. The algorithm also produces a perpendicular structure joining the original mesh with the offset mesh that consists of only planar elements (i.e. beams). 202_203

2 ACADIA 2015 COMPUTATIONAL ECOLOGIES 1 INTRODUCTION AND MOTIVATION Offset surfaces occur in a range of applications, from architectural settings, to modelling crystal growth (Figure 1), to dentistry methods, to name just a few. We focus on offsetting polyhedral surfaces (planar-faced mesh surfaces), specifically triangular meshes. There are several different types of offset surfaces, including vertex-, edge- or faceoffsets (Pottman et al, 2007) (Figure 2). In particular: a vertex-offset is a polyhedral surface whose vertices are at a constant non-zero distance from the corresponding vertices of the original mesh; an edge-offset is a polyhedral surface whose edges (more precisely, the lines containing the edges) are at a constant non-zero distance from the corresponding edges of the original mesh; a face-offset is a polyhedral surface whose faces (more precisely, the planes containing the faces) are at a constant non-zero distance from the faces of the original mesh. Another general strategy for offsetting is to preserve the shape of a mesh, but not preserve the combinatorial structure of the underlying mesh (the arrangement of vertices, edges, faces), see for example (Forsyth 1995; Jung et al 2004; Liu and Wang 2011). These methods are popular in settings that use a fine-grained triangular mesh in which the particular combinatorial details (vertices, edges, faces) of the mesh are secondary to the overall shape that the mesh is approximating. The offset surface may be composed of a mesh that is not related to the original by anything but shape. While vertex- and edge-offsets are useful in certain settings, neither approach will automatically preserve planar faces of polygonal meshes, and thus the resulting surface may not be a mesh. While vertex offsetting will automatically produce a mesh with planar faces for triangular meshes, the support beams (the distance between an edge and its offset) is unlikely to be planar, making it inappropriate for construction. In contrast, face-offsetting preserves planarity of faces by definition, and will also produce planar supports. In this paper we restrict our attention to the face-offsetting problem. The central issue for face-offsets is that the combinatorial structure of the original mesh is not preserved through this type of offsetting procedure. More concretely, only three planes meet generically in a vertex. Therefore, offsetting a vertex in a triangular mesh (which typically has an average of six incident faces) will not usually result in a single offset vertex with six incident faces, but rather the original vertex splits into a number of new vertices, each of which has three incident faces. It is possible to place restrictions on the original mesh to ensure that the combinatorial structure of the original mesh is preserved under face-offsetting, as described in Pottman and Wallner (2008) and Wang et al. (2007); however, these restrictions pose significant design challenges. In an architectural setting, face-offsetting is used to create layered surfaces (Figures 3 and 4). The details (face planes, intersection angles etc.) of the particular mesh are Figure 1 Crystals evolving over time toward their Euhedral planes. Photo: Geocosm LLC. a. c. Figure 2 Different types of offsets: (a) vertex-offset (b) face-offset (c) edge-offset b.

3 5.0 GEOMIMESIS/LANDFORMING HAMBLETON + ROSS Figure 3 Architectural details from Lassonde School of Engineering, York University, Toronto, Canada. ZAS Architects + Interiors. Photo: Dieter Janssen, March Figure 4 Lassonde building, detail. important, and the edges form visual elements in the resulting structures making it inappropriate to re-mesh the surface. Furthermore, architectural materials and assemblies have thickness, but concept design is usually done using two-dimensional digital surfaces. Making a digital design more real involves using geometric operations such as offsetting to achieve this thickness or depth. The problem is that the offset operation does not preserve many geometric properties easily achievable with thin surfaces (i.e. node coincidence where more than three planes meet, etc). In this paper we describe a face-offsetting procedure that pays attention to the particular structure of a mesh, but places no restriction on the geometry of the original mesh. We call this exact face-offsetting. Previous approaches to face-offsetting have involved restricting the input geometry. Pottmann and Wallner (2008) describe conditions on planar quadrilateral meshes that ensure the combinatorial structure of the offset mesh is identical to that of the original. Wang and Liu (2010) describe face-offsetting for hexagonal meshes which is another way of forcing a combinatorially equivalent face-offset for any offset distance. The method described in this paper places no restriction on the original geometry or combinatorics. Our algorithm does not force the face-offset mesh to be combinatorially equivalent, but is instead a method for computing the adjusted combinatorics of the offset mesh. This method includes the surfaces described in Pottman and Wallner (2008) and Wang and Liu (2010) as a subset. The result of our algorithm is an offset surface with reduced node complexity, but not at the cost of design intent. For instance, a triangular mesh with an average of six faces incident to every vertex will be offset to a mesh whose nodes typically have three incident faces. a. c. b. Figure 5 Given a triangle mesh, form the dual mesh using a vertexface correspondence: (a) mesh M; (b) forming dual; (c) dual mesh M/. 2 EXACT FACE-OFFSETTING 2.1 PROBLEM STATEMENT Given an (oriented two-manifold) polygonal mesh, find a valid (inward or outward) faceoffset mesh at an offset distance d. A mesh is valid if is also an oriented two-manifold polygonal mesh, without self-intersection. There are no restrictions on the geometry of the original surface other than face planarity. For triangular meshes, this is guaranteed. For quadrilateral or polygonal meshes, additional optimization may be necessary to achieve this initial condition. 2.2 PRELIMINARIES We use a vertex-face structure for a mesh M, and describe the combinatorial structure of the mesh by the incidence relationships of vertices on faces, together with the edges this implies (Figure 5a). The dual mesh M/ is the mesh formed from M by replacing each face of M with a vertex (located geometrically at the centroid of each face polygon) and connecting two vertices of M/ if the corresponding faces in M share an edge, see Figure 5. Similarly, faces in M/ correspond to vertices in M. 204_205

4 ACADIA 2015 COMPUTATIONAL ECOLOGIES Three planes in general position (no pair is parallel) meet at a point, and form a cone. Face-offsetting these planes will always result in a single intersection point, which moves along the cone axis determined by the three planes (Pottman and Wallner 2008) (three planes that intersect at a vertex are tangent to a unique circular cone). See also Figure 14. As a consequence, it is straightforward to describe the face-offset of a hexagonal mesh, as in Wang and Liu (2010), since each vertex has three incident faces, and this combinatorial structure is preserved under offsetting. If a vertex v 0 is incident to n planes, we say that v 0 is n-valent. After face-offsetting, the n planes will no longer generically intersect in a point (unless all of the planes surround a single cone axis as in Pottman and Wallner (2008)). Therefore, the face-offset of the planes at a particular vertex will possess new vertices and edges (but no new faces). The emergence of new vertices in the offset mesh corresponds to the creation of new faces in the dual mesh M/ (Alexandrov 2005, p ). In particular, the maximum possible number of new vertices is represented by a triangulation of the dual mesh M/ see Figure 6. Determining the face-offset mesh is therefore equivalent to finding a triangulation of the dual mesh. There are many possible triangulations of the dual mesh, which in turn correspond to the full range of possible face combinations at each offset vertex. A sample of these vertex types is shown in Figure 7. Note that this depicts combinatorial types rather than geometric types. Our task, therefore, is to find a triangulation that corresponds to a face-offset mesh that is both valid and geometrically optimal. To do so we introduce metrics on the quality of the offset mesh, namely a cost function that selects from the set of valid meshes to minimize the total edge-lengths and maximize the face-fairness. 2.3 EXAMPLE: VERTEX CONES To illustrate the approach we present the simplest non-trivial case, namely a 4-valent mesh vertex v 0, see Figure 8. The vertex v 0 corresponds to a quadrilateral face F v0 in the dual mesh, which is triangulated in one of two ways. That is, there are exactly two possible offset types at this vertex. a. c. e. Figure 6 Triangulations of the dual (a) and (d), and the corresponding offset topologies (c) and (f): (a) a triangulation of the dual mesh; (b) recovering offset original; (c) offset combinatorics; (d) a different triangulation of the dual mesh; (e) recovering offset; (f) offset corresponding to different triangulation. Figure 7 Detail of possible vertex types: some possible outcomes under face-offsetting for a vertex with ten incident faces. b. d. f. In Figures 9 and 10 we illustrate the two possible offset outcomes at a sample 4-valent vertex cone. In this case the vertex has one offset that is valid and another that is not. In other cases (for example when the vertex is hyperbolic (Wang et al 2007; Bobenko et al 2010)), there may be two valid offsets, one of which may be better than the other. 2.4 SKETCH OF ALGORITHM The algorithm has a local phase and a global phase. The local phase moves vertex by vertex over the mesh, and is appropriate for small offset distances. It is possible that edges or faces will vanish if the offset distance is sufficiently large. The global phase of the algorithm builds on the local phase to incorporate the changes to the combinatorics of the mesh that may occur with larger offset distances.

5 5.0 GEOMIMESIS/LANDFORMING HAMBLETON + ROSS a. b. We outline here the basic idea of the local phase of the algorithm. For a vertex v with n incident faces determining planes P 1,,P n : c. e. g. i. Figure 8 4-valent vertex cone (a) and corresponding quadrilateral face F v0 in the dual mesh (c). The face F v0 can be triangulated in one of two ways ((d) and (g)), which correspond to two possible offsets at v 0 ((f) and (i)): (a) 4-valent vertex v 0 ; (b) forming dual mesh; (c) quadrilateral dual face F v0 ; (d) first triangulation of dual; (e) recovering offset from triangulated dual; (f) first offset; (g) second triangulation of dual; (h) recovering offset from triangulated dual; (i) second offset. Figure 9 A sample 4-valent vertex (left) and the two candidate o sets (middle, right). One candidate offset contains two selfintersecting faces (right) and is therefore not a valid mesh. d. f. h. 1. For every triple of planes {P i,p j,p k } compute the cone axis defined by these planes. There will be (n/2) combinatorially distinct axes, but note they need not be geometrically distinct. 2. Compute the cloud of candidate offset vertices using the offset distance d to determine exactly one candidate vertex along each cone axis. 3. Compute all possible triangulations of the dual mesh face corresponding to v. 4. Recover the local offset mesh from the triangulated face, using the vertex positions from Step 2 (the candidate offset vertices are indexed by triples corresponding to the vertices of the triangles in the dual face). Discard any triangulations which generate self-intersecting faces. 5. Compute the cost of the local geometry based on the lengths of the new edges and the fairness of faces. 6. Return the triangulation with the lowest cost. Repeat for each vertex until the dual mesh is completely triangulated. Finally, recover the final offset from the triangulated dual mesh. It is possible that under offsetting, certain edges or eventually faces may disappear. This is a more significant combinatorial change that is has not been addressed by the algorithm above. However, a method for detecting this global combinatorial change is available and will be recorded in a future paper. 3 RESULTS Figure 11 and Figure 12 illustrate our algorithm at work. Figure 11 shows the working surface with a triangular mesh structure. The vertices have between four and six incident faces (our method can also accommodate higher valence vertices). The surface mesh is offset to the surface shown in Figure 12. Here planar structural supports meet at generalized vertices, which are actually several closely placed 3-valent vertices. For comparison, see Figure 13; see also Figure 14 which shows three cone axes originating in a single node and splitting into three new vertices in the offset surface. Figure 10 Perspective view of the 4-valent vertex and the two candidate offsets of Figure 9. The original vertex is shown in the back, with the two candidate offset surfaces to the right.. 4 APPLICATIONS AND FURTHER WORK There are several immediate extensions of this work. No part of this algorithm depends on choosing a single offset distance to apply to each face. Therefore, it is possible to assign a unique offset distance to each face or groupings of faces. For example, varying offset distance would allow the user to incorporate distance or angle constraints (e.g. avoiding structural elements in a design). There may also be different decision criteria required for particular applications. Our method allows a user to flip through the range of possible valid offsets of a mesh, or to specify a decision metric for automatic mesh selection. 206_207

6 ACADIA 2015 COMPUTATIONAL ECOLOGIES There are some interesting related questions for future investigation. Edge-offset models are desirable because they provide a simplified construction method with structural beams of uniform size. Under what conditions can we use our face-offset method with variable distances to produce an edge-offset? That is, is there a set of offset distances for the faces of a mesh such that the resulting face-offset is also an edge-offset? While this is, in general, an over-constrained geometric problem, there may be a restricted class for meshes for which a solution is available. Figure 11 Triangular mesh design surface. The offset surface is shown in Figure 12. Figure 12 The offset surface resulting from offsetting the mesh in Figure 11. The faces incident to a vertex are no longer coincident in a point. Contrast Figure 13. Figure 13 Actual offset vertex using ad hoc methods. Lassonde building, detail. Figure 14 Detail of the offset surface of Figure 12 with cone axes identified (red). The axes intersect the original vertex, then fan out into the resulting offset vertices. There is a fourth cone axis that has been omitted for visual clarity.

7 5.0 GEOMIMESIS/LANDFORMING HAMBLETON + ROSS REFERENCES Alexandrov, A. D. (2005). Convex Polyhedra. Springer Monographs in Mathematics, tr. N. S. Dairbekov, S. S. Kutateladze and A. B. Sossinsky. Springer-Verlag, Berlin. Bobenko, A. I., H. Pottmann, and J. Wallner. (2010). A Curvature Theory for Discrete Surfaces Based on Mesh Parallelity. Mathematische Annalen, 348(1): Forsyth, M. (1995). Shelling and offsetting bodies. In Proceedings of the third ACM symposium on Solid modeling and applications, eds. C. Hoffmann and J. Rossignac, New York: ACM. DOI / Jung, W., H. Shin, and B. K. Choi. (2004). Self-intersection Removal in Triangular Mesh Offsetting. Computer- Aided Design and Applications, 1(1-4): Liu, S. and C. C. Wang. (2011). Fast Intersection-free Offset Surface Generation from Freeform Models with Triangular Meshes. IEEE Transactions on Automation Science and Engineering 8(2): Pottmann, H., A. Asperl, M. Hofer, and A. Kilian. (2007). Architectural Geometry. Exton, Pennsylvania: Bentley Institute Press. Pottman, H. and J. Wallner. (2008). The Focal Geometry of Circular and Conical Meshes. Advances in Computational Mathematics 29(3): Wang, W., J. Wallner, and Y. Liu. (2007). An Angle Criterion for Conical Mesh Vertices. Journal for Geometry and Graphics 11(2): Wang, W. and Y. Liu. (2010). A Note on Planar Hexagonal Meshes. In The IMA Volumes in Mathematics and Its Applications, Volume 151: Nonlinear Computational Geometry. eds. I.Z. Emiris, F. Sottile, T. Theobald, New York: Springer-Verlag. 208_209

8 ACADIA 2015 COMPUTATIONAL ECOLOGIES DANIEL HAMBLETON Daniel Hambleton is the Director of MESH Consultants Inc., a Toronto-based consulting firm that offers Applied Mathematics and Development services to the Digital Design Industry. He has worked extensively across a variety of markets, such as: architecture, product design, energy, software development, and engineering. Although his research is focused on computational geometry and physics simulations, he has extensive experience with interdisciplinary projects and unique collaborations. ELISSA ROSS Elissa Ross holds a PhD in mathematics from York University (Toronto) where her research focused on the rigidity and flexibility of periodic (repetitive) structures. She has additional expertise in computational geometry, graph theory and tilings/ patterns, and a long history of collaborative and interdisciplinary projects. At MESH Consultants Inc., Elissa conducts research in architectural geometry, adds to the breadth of the geometry consulting services, and develops in-house tools for 3D geometry applications.