Meshing Skin Surfaces with Certified Topology

Transcription

1 Meshing Skin Surfaces with Certified Topology Nico Kruithof and Gert Vegter Department of Mathematics and Computing Science University of Groningen, The Netherlands Abstract We present an algorithm that approximates a skin surface with a topologically correct mesh. The number of vertices of the mesh is quadratic in the number of input balls defining the skin surface. We describe two refinement algorithms to improve the quality of the mesh and show that the same algorithm can also be used for meshing a union of balls. 1 Introduction Skin surfaces, introduced by Edelsbrunner in [9], have a rich combinatorial structure that makes them suitable for modeling large molecules in biological computing. Meshing such surfaces is often required for further processing of their geometry, like in numerical simulation and visualization. We present an algorithm for meshing skin surfaces with guaranteed topology. Large molecules can be modeled using skin surfaces by representing each atom by a sphere. Atoms that lie close to each other are connected by quadratic patches. These patches make the transition between the atoms tangent continuous. A skin surface is parameterized by a set of weighted points (input balls) and a shrink factor. If the shrink factor is equal to one, the surface is just the boundary of the union of the input balls. If the shrink factor decreases, the skin surface becomes tangent continuous, due Partially supported by the IST Programme of the EU as a Shared-cost RTD (FET Open) Project under Contract No IST (ECG - Effective Computational Geometry for Curves and Surfaces). P.O. Box 800, 9700 AV Groningen, The Netherlands to the appearance of patches of spheres and hyperboloids connecting the balls. In [14] we construct an algorithm that approximates an arbitrary surface with a skin surface. The approximation is homeomorphic to the skin surface and the Hausdorff distance between the two surfaces is arbitrarily small. Two surfaces embedded in three space are isotopic if there is a continuous deformation within the embedding space that transforms one surface into the other one. In particular, the surfaces are homeomorphic. The algorithm presented in this paper constructs a mesh isotopic to the skin surface in two steps: it constructs a coarse, isotopic mesh which it subsequently improves. The complexity of the coarse mesh is quadratic in the number of input balls, and is independent of the shrink factor. This is worst case optimal. For the second step a broad range of refinement algorithms can be used. The algorithms have to be adapted slightly in order to ensure the isotopy. We show how this is done for the refinement algorithms in [13] and [7]. The sqrt-3 subdivision algorithm [13] is very fast, and refines the size of the triangles. However, it does not improve the quality of the mesh elements in terms of angle size. The refinement step preserves the isotopy property. Chew s algorithm [7] is another refinement method which improves the quality of the mesh in terms of the angles and size of the triangles. The quality mesh is suitable for numerical computations. Related work. Most existing algorithms meshing implicit surfaces do not guarantee topological equivalence of the surface and the mesh constructed. The marching cubes algorithm [16] subdivides a region into cubes 1

2 and triangulates the surface within these cubes based on whether the vertices of the cube lie inside or outside the cube. A variant of this algorithm that follows the surface is [3]. The marching triangulation method [11] extends a small initial mesh by walking over the implicit surface, starting from a seed point. Our paper [15] presents a marching triangulation method for meshing skin surfaces by carefully choosing the step size during the walk over the mesh. However, as the shrink factor goes to one or to zero, the size of the mesh goes to infinity. The algorithm presented in [4] is among the first general methods guaranteeing topological equivalence of the implicit surface and the mesh. The algorithm in [6] constructs a topologically correct mesh approximating a skin surface in the special case of a shrink factor 0.5. It is likely that this algorithm can be generalized to work for arbitrary shrink factors, but this would probably result in a denser mesh in order to guarantee the topology. Another method for visualizing molecules uses Molecular Surfaces [8]. Visualization algorithms for this type of surfaces are presented in [1, 2]. Outline. In Section 2 we review the theory of skin surfaces as presented in [9] and describe the method to subdivide these surfaces into patches of degree two (patches of hyperboloids and balls). Section 3 describes the construction of the coarse mesh. We also construct an isotopy between this mesh and the skin surface. In Section 4, we describe two methods to refine and improve the coarse mesh. We show in Section 5 how the algorithm can be adapted so that it meshes the union of a set of balls. 2 Skin surfaces This section is a summary of [9]. For further details on skin surfaces we refer to this paper. A skin surface is defined in terms of a finite set of balls (weighted points) and a shrink factor s, with 0 < s 1. For s = 1, the skin surface is the boundary of the union of balls. For smaller s, the radii of the balls in the input set are shrunk, and patches of spheres and hyperboloids are generated between adjacent balls, making the skin surface tangent continuous. All definitions are given for skin surfaces in 3D, but can easily be transformed into the definition of skin curves in 2D. All figures show skin curves in the plane. The space of weighted points. A 3D weighted point ˆp is a pair ˆp = (p, p) with center p R 3 and weight p R. A ball with center p and radius R corresponds to a weighted point (p, R 2 ). This definition is slightly different from [9], in the sense that there a weighted point is associated with the sphere bounding the ball. On the set of weighted points we define a pseudo-distance function: π(ˆp, ˆq) = p q 2 p q, (1) for ˆp = (p, p) and ˆq = (q, q). We will call a point with zero weight an unweighted point. The distance from a weighted point ˆp to an unweighted point x follows from Equation (1) by taking q = x and q = 0. All unweighted points with nonpositive distance to ˆp form the ball with center p and radius p. We will make no distinction between the weighted point and this ball. Two points with zero distance are by definition orthogonal. An orthosphere of a set of weighted points P is, by definition, a sphere orthogonal to each of the weighted points in P. General position. In the remainder of this paper we assume general position, by which we mean that no 5 weighted points are equidistant to a point in IR 3 and no k+2 centers of weighted points lie on a common k-flat for k = 0, 1, 2. Several methods like [10] exist to symbolically perturb a data set and ensure these conditions. Note that, under this genericity condition, an orthosphere only exists if P 4. The space of weighted points inherits a vector space structure from R 4 via the bijective map Π : R 3 R R 4, defined by Π(ˆp) = (ξ 1, ξ 2, ξ 3, p 2 p), with p = (ξ 1, ξ 2, ξ 3 ). Addition of two weighted points and the multiplication of a weighted point by a scalar are defined in the vector space inherited under Π. Combining these two operators, we introduce the notion of shrinking and growing of weighted points. Given a shrink factor s R, we associate with a weighted point ˆp the shrunk 2

3 Figure 1: The skin curve of two weighted points (the two larger circles). The smaller circles form a subset of the shrunk convex hull of the input points. The boundary of the shrunk convex hull forms the skin curve. weighted point ˆp s defined as follows. Let ˆp = (p, 0). Then ˆp s = s ˆp + (1 s) ˆp. A simple calculation shows that ˆp s = (p, s p). If P is a set of weighted points, we denote by P s the set obtained by shrinking every point of P by a factor s. The body of a skin surface bdy s P and skin surface skn s P, associated with a set P of weighted points, are defined by bdy s P = (conv P) s (2) skn s P = bdy s P. (3) The weighted Voronoi diagram. The weighted Voronoi cell, Voronoi cell for short, of a weighted point ˆp P consists of all points that are closer with respect to the pseudo-distance π defined in Equation (1) to ˆp than to any other point in P. Formally: Here conv (P) R 3 R is the convex hull with respect to the vector space structure inherited under Π of a set of weighted points P, whereas denotes the boundary in three space of the union of the set of balls. For a skin curve in 2D associated with two weighted points: see Figure 1. One of the main features of skin surfaces is that they can be effectively subdivided into patches of degree two, more precisely, into parts of spheres and hyperboloids. Later in this section we will show how this is done. To determine these patches, we introduce the mixed complex, which subdivides the skin surface into patches of degree two. The mixed complex is an intermediate structure between the weighted Delaunay triangulation (or: regular triangulation) and its dual, viz. the weighted Voronoi diagram (or: the power diagram). Vˆp = {x R 3 π(ˆp, x) π(ˆq, x), for all ˆq P}. A weighted Voronoi cell is a convex polyhedral region, possibly unbounded, or empty. Furthermore, a weighted Voronoi cell Vˆp associated with the weighted point ˆp = (p, p) does not necessarily contain p. With a subset X P we associate the region ν X defined by ν X = ˆp X Vˆp which we will call a weighted Voronoi l-cell (l = dim ν X = 4 X ), if it is not empty and X is in general position. Then it also follows that X 4. In the non-degenerate case, every 0-cell is a point, which is the intersection of four adjacent Voronoi cells. A 1-cell is a line segment, possibly unbounded. And a facet (2-cell) is the intersection of the boundary of two 3-cells. The 2-cell is a subset of the set of unweighted points equidistant to the two weighted points associated with the 3-cells. A 3-cell is a nonempty Voronoi-cell of some ˆp P. The weighted Voronoi diagram is the subdivision of 3-space generated by the non-empty Voronoi cells: Vor P = {ν X X P ν X }. The weighted Delaunay triangulation. The weighted Delaunay triangulation, is dual to the weighted Voronoi diagram, and is defined in terms of weighted Delaunay simplices. A weighted Delaunay 3 l-cell exists for every non-empty weighted Voronoi l-cell ν X Vor P and is the convex hull of the centers of the points in X. Denoting this cell by δ X, we have δ X = conv {p ˆp X }. The weighted Delaunay triangulation is the subdivision generated by these cells: Del P = {δ X ν X Vor P}. We use the following property of the Delaunay triangulation and the Voronoi diagram: Observation 1 A Delaunay cell and its dual Voronoi cell span orthogonal flats. Note that δ X and ν X can be disjoint, but their affine hulls always intersect in one point. We will call this point the focus of X. Let f be the focus of X. Since the focus lies on the affine hull of the weighted Voronoi cell, it follows that π(ˆp, f) = π(ˆq, f) for all ˆp, ˆq X. For further reading on the space of circles we refer to [17]. The mixed complex. The mixed complex, associated with a scalar s [0, 1], is obtained by taking Minkowski sums of shrunk cells of 3

4 the weighted Delaunay triangulation and the weighted Voronoi diagram. More precisely, for every X P with ν X a mixed cell is defined by ˆp 0 µ s {ˆp 0,ˆp 1 } µ s {ˆp 1 },{ˆp 0,ˆp 1 } µ s X = s ν X (1 s) δ X, µ s {ˆp 1 } where denotes the multiplication of a set by a scalar and denotes the Minkowski sum. For s = 0 the mixed cell is the Delaunay cell. When s increases it deforms affinely into the Voronoi cell for s = 1. We will call a mixed cell corresponding to a Delaunay l-cell a mixed cell of type l. Since both a weighted Delaunay cell and its dual weighted Voronoi cell are convex polyhedra, it follows that mixed cells are also convex polyhedra. For X = 1 the corresponding mixed cell (a mixed cell of type 0) is a convex polyhedron, i.e., the corresponding Voronoi 3-cell shrunk towards the corresponding Delaunay vertex with a factor s. A mixed cell of type 1 is a prism with the shrunk Voronoi facet as its base. Similarly, a mixed cell of type 2 is a prism with the shrunk Delaunay triangle as its base. A mixed cell of type 3 corresponds to a Delaunay tetrahedron shrunk towards the Voronoi vertex. The mixed complex, associated with a shrink factor s, consists of the set of all mixed cells: Mix s P = {µ s X ν X Vor P}. It is a partition of 3-space into convex polyhedral cells. The mixed cells and their intersections form a polyhedral complex. To be more precise, for two sets X, X P, such that δ X, δ X Del P, we define a polyhedral cell µ s X,X as: µ s X,X = µs X µ s X. Every polyhedral cell µ has a unique pair of labels X, X such that X X and µ = µ s X,X and has dimension 3 + X X, see Table 1. The following lemma results from the definition of a polyhedral cell. Lemma 2 µ s X,X = s ν X (1 s) δ X, for X X. We omit the proof from this version. Quadratic patches of a skin surface. The body of a skin surface is by definition the union of the shrunk convex hull of its weighted points v ˆp 1 µ s {ˆp 0,ˆp 1,ˆp 2 } ˆp 2 Figure 2: The skin curve of four weighted points (the dotted circles). Each mixed cell contains parts of an hyperbola or a circle. Some labels of mixed cells are given. Note that v = µ s {ˆp 0,ˆp 2},{ˆp 1,ˆp 2} = µ s {ˆp. 2},{ˆp 0,ˆp 1,ˆp 2} P, see Equation (2), which we rewrite to the following: bdy s P = {(convx ) s X P}. In [9] Edelsbrunner shows that only the subsets X for which ν X can touch the boundary of the skin surface. The other sets X with ν X = are contained inside these subsets. As a result it follows that each part of the skin surface is generated by the shrunk convex hull of at most 4 points. Furthermore, a subset X with ν X generates exactly the part of the skin surface that lies inside the mixed cell µ s X. A two-dimensional example is given in Figure 2. The dotted circles denote the weighted points in the input set. All rectangles (mixed cells corresponding to the Delaunay edges) contain hyperbolic patches. These hyperbolic patches are the boundary of the union of the shrunk convex hull of the two vertices of the Delaunay edge. All other cells contain circular arcs. Depending on whether the mixed cell corresponds to a Delaunay vertex or a Delaunay face, the interior of the skin curve lies inside or outside the circle. In 3D the mixed complex divides the skin surface into parts of degree two in a similar way. Assume we have a coordinate system x 1, x 2, x 3 such that the affine hull through a weighted Delaunay k-cell δ X is aligned along the first k axes and its dual weighted Voronoi 3 k-cell is aligned along the other axes. Then the focus lies at the origin and δ X skn s P = δ X S, 4

5 X \ X polyhedron polygon line segment point 2 prism rectangle line segment 3 prism triangle 4 tetrahedron Table 1: The shape of mixed polyhedra µ s X,X for s (0, 1) where S is the zero set of the function I X with: I X (x) = 1 1 s k x 2 i + 1 s i=1 3 i=k+1 x 2 i R 2. (4) where x = (x 1, x 2, x 3 ) and R 2 is the pseudodistance between the focus and a weighted point corresponding to the vertex of the Delaunay cell. Let k = X 1. For k = 0 and k = 3, the mixed cell clips a sphere. These cases are distinguished by the fact that, for k = 0, the sphere is convex (it contains the body), whereas for k = 3, it is concave. For k = 0, the mixed cell corresponds to a weighted Delaunay vertex. These mixed cells clip a shrunk weighted point from the input set. For k = 1 and k = 2 the mixed cell clips a hyperboloid. For k = 1 the patch lies on a hyperboloid with symmetry axis aligned along the Delaunay edge and for k = 2 it is a patch of a hyperboloid with symmetry axis aligned along the Voronoi edge. These patches of hyperboloids form a tangent continuous transition between the spherical patches. For the proof that these patches form the corresponding skin surface, we refer to [9]. The following observation follows from the tangent continuity of a skin surface. Observation 3 The restrictions of I X and I X and their gradients to µ s X,X are identical. The center. For the construction of the mesh we will use the center of a polyhedral cell. First we show a property of the focus. Lemma 4 The points in µ s X,X closest to foc(x ) and foc(x ) are equal, for X X. Proof. By definition of the focus, foc(x ) and foc(x ) lie on F = aff(ν X ) aff(δ X ), which is a k-flat, with k = X X. Lemma 2 says that µ s X,X = s ν X (1 s) δ X, which is orthogonal to F. Thus the projections of foc(x ) and foc(x ) on the affine hull of µ s X,X are equal and therefore the points of µ s X,X closest to foc(x ) and foc(x ) also coincide. The center c(µ s X,X ) is the point in µ s X,X closest to foc(x ). The previous lemma states that this definition is independent of the focus. In the construction of the mesh we use the fact that a mixed polyhedron is star shaped with respect to its center. 3 Meshing skin surfaces In this section we define the approximating triangulation and construct an isotopy between the triangulation and the skin surface. In order to do this, we construct an infinite set of open, disjoint line segments, called transversal segments, such that each point on the skin surface lies on a unique transversal segment and each transversal segment intersects the skin surface exactly once. Then we construct the approximating mesh in such a way that it also has these two properties with regard to the transversal segments. In the remainder of this section we construct the transversal segments, define the mesh and show the isotopy. Then we show that the complexity of the mesh is quadratic in the number of input balls. 3.1 Parameterizing the skin surface To define the transversal segments, we subdivide the skin surface by intersecting it with a simplicial complex. A simplicial complex is the collection of faces of a finite number of simplices, any two of which are either disjoint or meet in a common face [18]. A tetrahedral complex is a simplicial complex for which the maximal dimension of the cells is three. Recall that 5

6 c e15 c v1 c v2 c e12 c v1 c v5 c f c e45 c e23 c v2 = c e12 (a) Circular facets c e24 c f c e13 (b) Hyperbolic facets c v4 c e34 c v4 c e34 c v3 c v3 Figure 3: Splitting the facets of a mixed cell. The points c v, c e and c f are the centers of a vertex, an edge and a facet respectively and are labeled by their indices. the mixed complex subdivides the skin surface into patches of quadrics. We assume that there is a large bounding box around the skin surface making all mixed cells finite. We subdivide each polyhedral cell of the mixed complex into tetrahedra and construct the transversal segments within each tetrahedron. The vertices of the tetrahedral complex are centers of polyhedral cells. First we triangulate the boundary of the mixed cell as described presently and then construct tetrahedra by taking the join of each triangle with a point inside the mixed cell. Triangulation of facets. Each mixed cell µ s X contains either a part of a sphere or a hyperboloid, this part being the zero set of I X. We distinguish two types of facets: circular facets are facets for which the contour lines of I X on the facet are circles. The other facets are called hyperbolic because the contour lines are hyperbolas on the facet. Observation 3 states that all facets inherit the same label from both incident mixed cells (either as circular or hyperbolic). All facets of mixed cells of type 0 and 3 are spherical. The facets of a mixed cell of type 1 or 2 are spherical if they touch a mixed cell of type 0 or 3, and hyperbolic if they touch a mixed cell of type 1 or 2. We triangulate circular and hyperbolic facets differently. Circular facets are triangulated by splitting each edge at its center and then creating a star of triangles around the center of the facet towards each split edge, as depicted in Figure 3(a). Hyperbolic facets are rectangles with their edges parallel or perpendicular to the symmetry axis of the corresponding hyperboloid. Similar to the circular case we split the edges of the facet at their centers. Then we subdivide the facet in the symmetry plane of the hyperboloid, if it intersects the symmetry plane. For each non-empty subdivided facet we construct a star of triangles. This star is created from the center of the edge parallel to the symmetry plane to the split edges in the subdivided facet, see Figure 3(b). We discard triangles for which two or more vertices coincide (for example triangle c v2, c e12, c f in Figure 3(a)). Lemma 5 The critical point of I X, restricted to a triangle of the decomposition of a facet of a mixed cell µ s X lies at vertices of the triangle. Proof. The critical point of I X is the projection of the focus of the mixed cell on the plane through the facet. If the facet contains this point, then it is, by definition, the center of the facet. Otherwise I X restricted to the triangle has no critical points. A similar argument applies for the edges of the facet. Thus, all critical points lie on vertices of triangles. Construction of tetrahedra. Mixed cells of type 0 and 3 are decomposed into tetrahedra by taking the join of each triangle on each facet and the center of the mixed cell. Again we discard tetrahedra for which two vertices coincide. For the decomposition of a mixed cell of type 1 or 2, we split the mixed cell in the symmetry plane of the hyperboloid defined in the mixed cell. In this way we obtain two prisms if the mixed cell intersects the symmetry plane and one otherwise. Then we triangulate the boundary facets of a prism and construct tetrahe- 6

7 Figure 4: The three different configurations of a tetrahedron. White and black vertices lie on different sides of the skin surface. dra by taking the join of each triangle with the center of the facet farthest away from the symmetry plane. Lemma 6 The cells of all tetrahedra form a simplicial complex. Proof. This is trivially true for two simplices within the same mixed cell. For two simplices in different mixed cells, the simplices are either disjoint or touch on a common face of the mixed cells. This face is triangulated in the same way in both cells since it inherits the same label from both incident mixed cells and the centers are identical (Lemma 4). Construction of transversal segments. A tetrahedron τ in the tetrahedral complex is active if the skin surface intersects the tetrahedron. Let τ be an active tetrahedron. If we assume that the simplicial complex is in general position, by which we mean that the skin surface does not contain the vertices of the simplicial complex, then the vertices of τ can be categorized into two groups. The set Vτ contains the vertices inside and V τ + the vertices outside the skin surface and let W τ ± = conv (V τ ± ). Definition 7 For an active tetrahedron τ, a transversal segment is an open line segment (w τ, w + τ ), for w ± τ W ± τ. Lemma 8 A transversal segment intersects the skin surface transversally in a unique point. Conversely, each point inside an active tetrahedron lies on a transversal segment and, therefore, also each point on the skin surface within the tetrahedron. We omit the proof from this version, but remark that it follows from the fact that I X is strictly monotone along transversal segments. The mesh. We mesh the skin surface within the tetrahedra. Based on whether the vertices of the tetrahedron lie inside or outside the skin surface we can distinguish three cases as depicted in Figure 4. We place the vertices of the triangles on the intersection points of the edges with the skin surface. The third configuration remains ambiguous, since the common edge of the two triangles can be flipped. In fact, both configurations are isotopic, since they intersect each transversal segment once and they have the same border edges. Lemma 9 The skin surface and the approximating mesh are isotopic. Again, we omit the trivial proof from this version. 3.2 Complexity analysis We show the following result. Lemma 10 For an input set with n balls, the complexity of the mesh O(n 2 ) and this is optimal. The upper bound is obtained as follows. For n balls the complexity of the Delaunay triangulation is O(n 2 ), see [5]. The size of the mixed complex is linear in the size of the Delaunay triangulation, since the complexity of a mixed cell is linear in the complexity of the Delaunay and Voronoi cell it is constructed from. The number of tetrahedra within a mixed cell is linear in the complexity of the mixed cell and within each tetrahedron we construct at most two triangles. Thus, the mesh is linear in the size of the Delaunay triangulation, which is O(n 2 ). To show that this is optimal, we use the standard way of creating a Voronoi diagram with quadratic complexity. The union of these balls has O(n 2 ) holes, and therefore the skin surface has O(n 2 ) holes. Any triangulation of such a surface has Ω(n 2 ) size. 4 Mesh refinement The topologically correct mesh, as defined in Section 3 is rather coarse and may contain long 7

8 Figure 5: The Sqrt-3 subdivision method applied twice. Left: the original triangle, in the middle the subdivided triangle. On the right are the vertices placed on the skin surface. and skinny triangles. Therefore, we present two mesh refinement algorithms to improve the mesh. Again, we require that these algorithms produce a mesh that is isotopic to the skin surface. A mesh M is valid if it satisfies the following properties: 1. each transversal segment intersects the M once and 2. each point of M lies on a transversal segment. These conditions guarantee that the isotopy defined in Lemma 9 remains valid for the refined mesh. Changing the mesh. The refinement algorithms we consider in the remainder of this section are iterative. Like in many refinement methods, each refinement step can be decomposed into three phases: change the structure of the mesh, move vertices and adjust the structure of the mesh again. E.g., an edge flip can be decomposed into splitting the edge in a common intersection point with respect to some parameterization, moving the new vertex towards the other edge and removing the added vertex again. The first and last step do not change the embedding of the mesh in three-space, but only modify its structure. On the other hand they ensure that property (1) is satisfied as long as we deform the mesh by moving points along the transversal segments. Given the changing simplices, it is easy to test for property (2) once we know the tetrahedra containing them. 4.1 Sqrt-3 method In this section we show that for the sqrt-3 subdivision method [13] conditions (1) and (2) are satisfied (if we apply the subdivision step twice). The main observation is that the subdivided triangles lie in the same tetrahedron as the coarse triangle. This is the case since the edges of the coarse triangles are retained in the subdivision step. The subdivision algorithm then moves each vertex along the transversal segments towards the skin surface. Since the end points of these transversal segments lie on different sides of the skin surface, the vertices are replaced within the tetrahedron and the isotopy can be established. The mesh converges to the skin surface under the sqrt-3 subdivision algorithm. Moreover, properties (1) and (2) described in the beginning of this section hold automatically, which makes the algorithm fast. On the other hand, the subdivision algorithm does not improve the quality of the triangles. Therefore this method is not suitable for constructing a mesh for numerical simulations. 4.2 Chew s algorithm In this section we use Chew s algorithm [7] to improve the quality of the triangles of the coarse mesh. The quality of a triangle is measured by its shape and its size. Well-shaped triangles have angles between 30 and 120 degrees. What well-sized means is defined by the user, e.g., the size of a triangle can be taken proportional to the inverse of the curvature of the surface. Chew s algorithm maintains and improves a Delaunay triangulation on the skin surface. The definition of a Delaunay triangulation in 2D does not extend directly to the mesh. In fact, the empty circle property is adapted as follows. Definition 11 Let ab be an edge of the mesh belonging to two triangles abc and abd. The sphere σ is the sphere through a, b and c, centered on the surface. The edge ab is locally Delaunay, if d lies outside σ. Lemma 5 in [7] guarantees that the empty circle test gives consistent results if we exchange abc and abd. Since a skin surface is closed and we do not constrain edges, the algorithm greatly simplifies. It boils down to two steps. 8

9 1. Construct a Delaunay triangulation from the initial mesh using edge flips. 2. Add the circumcenter of the triangle with largest circumcircle that violates the quality criterion as a vertex of the Delaunay triangulation, until no more violating triangles. The algorithm has two primitive operations: edge flips and the insertion of new vertices into the mesh. Edge flips. Consider two triangles with a common edge, say abc and abd, for which we want to flip ab into cd. Conceptually this is done as follows: 1. Find the transversal segment l that intersects both ab and cd. 2. Split ab in the intersection point e with l and add edges ce and de. 3. Move e along l towards cd 4. Remove ae, be and e. In the rare case that the line segment l does not exist, flipping the edge would violate property (1) (the mesh would intersect a transversal segment more than once). Therefore we do not flip the edge. If, in step 3 the mesh moves outside the union of the tetrahedra, then property (2) is violated. To prevent this from happening we insert the first point that would lie outside the union of the tetrahedra into the mesh. Then we move it towards the skin surface. Vertex insertion. Let x be a point we want to insert into the mesh. Let l be the transversal segment containing x and abc the triangle that intersects l. We split abc in the intersection point of l and abc and move this point towards the skin surface. Again we have to check property (2). If property (2) fails, we add a vertex to the mesh on the point where it is first violated. 5 Union of balls The algorithm can also be used to mesh the union of a set of balls. For a shrink factor one, the skin surface of a set of balls is the union of these balls. In this case, the mixed complex is the Voronoi diagram. This means that only mixed cells of type 0 are three-dimensional cell. This greatly simplifies the set of tetrahedra. We also want to retain edges of the mesh on the intersection of two balls. The subdivision algorithm ensures this by construction. Chew s algorithm has to be extended to allow constrained edges, see [7]. 6 Conclusion and future work. We presented an algorithm that constructs a mesh that is isotopic to the skin surface and discussed two methods to refine this mesh. We want to analyze other mesh optimization techniques like [12]. The algorithm we presented is static in the sense that it generates a mesh for a fixed set of input balls. The rigid construction of this mesh makes us believe that it is also possible to maintain the coarse mesh while deforming the input set. References [1] C. Bajaj, H. Y. Lee, R. Merkert, and V. Pascucci. Nurbs based b-rep models for macromolecules and their properties. In Proceedings of the fourth ACM symposium on Solid modeling and applications, pages ACM Press, [2] C. L. Bajaja, V. Pascuccia, A. Shamira, R. J. Holt, and A. N. Netravali. Dynamic maintenance and visualization of molecular surfaces. Discrete Applied Mathematics, 127(1):23 51, April [3] J. Bloomenthal. Polygonization of implicit surfaces. Comput. Aided Geom. Design, 5(4): , [4] J.-D. Boissonnat, D. Cohen-Steiner, and G. Vegter. Meshing implicit surfaces with certified topology. Technical report, IN- RIA, Sophia Antipolis, September [5] J.-D. Boissonnat and M. Yvinec. Algorithmic Geometry. Cambridge University Press, UK, Translated by Hervé Brönnimann. [6] H.-L. Cheng, T. K. Dey, H. Edelsbrunner, and J. Sullivan. Dynamic skin triangu- 9

10 Figure 6: A skin surface, the coarse mesh, the mesh after one subdivision step and Chew s algorithm applied to the mesh. Figure 7: The coarse mesh of a molecule from the front and from the side. lation. Discrete Comput. Geom., 25: , [7] L. Chew. Guaranteed quality Delaunay meshing in 3d. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages , [8] M. L. Connolly. Analytical molecular surface calculation. Journal of Applied Crystallography, 16(5): , Oct [9] H. Edelsbrunner. Deformable smooth surface design. Discrete Comput. Geom., 21:87 115, [10] H. Edelsbrunner and E. P. Mücke. Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms. ACM Trans. Graph., 9(1):66 104, [11] E. Hartmann. A marching method for the triangulation of surfaces. Visual Comput., 14(3):95 108, [12] H. Hoppe, T. DeRose, T. Duchamp, J. Mc- Donald, and W. Stuetzle. Mesh optimization. In Proceedings of the 20th annual conference on Computer graphics and interactive techniques, pages ACM Press, [13] L. Kobbelt. 3 -subdivision. In Proceedings of the 27th annual conference on Computer graphics and interactive techniques, pages ACM Press/Addison- Wesley Publishing Co., [14] N. Kruithof and G. Vegter. Approximation by skin surfaces. In 8th ACM Symposium on Solid Modeling and Applications, 2003, pages 86 95, [15] N. Kruithof and G. Vegter. Triangulating skin surfaces. Technical Report ECG- TR , Rijksuniversiteit Groningen, [16] W. Lorensen and H. Cline. Marching cubes: a high resolution 3d surface construction algorithm. Comput. Graph., 21(4): , [17] D. Pedoe. Geometry, a comprehensive course. Dover Publications, New York, [18] G. Vegter. Computational topology. In J. E. Goodman and J. O Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 28, pages CRC Press LLC, Boca Raton, FL,

11 Figure 8: The caffeine molecule. Pictures are in the same order as Figure 6. We do not show the edges in the bottom row because the pictures would get to crowded. 11