Local Surface Approximation and Its Application to Smoothing in Three-Dimensional Indirect Mesh Generation

Local Surface Approximation and its Application to Smoothing in Three-dimensional Indirect Mesh Generation

Dorit Merhof a , Roberto Grosso a, Udo Tremel b G¨unther Greiner a, aComputer Graphics Group, Friedrich-Alexander University, Erlangen, Germany bEADS Military Aircraft, Munich, Germany

Abstract

Smoothing techniques are of major importance for the generation of surface meshes. For this reason a considerable amount of research has been spent on developing a large variety of sophisticated smoothing approaches. However, these methods either require direct access to the analytic surface description or are restricted to flat meshes in two dimensions. In particular, if no analytic surface data is available as in the case of indirect mesh generation, it is not possible to reposition nodes on a three-dimensional geometry. To cope with this problem, this paper presents an approach based on the local approximation of an analytic surface which is used for repositioning nodes when smoothing is performed. Furthermore, the paper shows how established two-dimensional smoothing techniques such as length smoothing, angle smoothing and isoparametric smoothing are adapted to three dimensions by combining them with the local surface approximation. A quadrilateral mesh generator that implements the presented techniques finally demonstrates the benefits of this approach.

Key words: local surface approximation, smoothing, three-dimensional surface mesh, indirect mesh generation, quadrilateral

1 Introduction

Nowadays, the generation of surface meshes is a common engineering problem. The surface to be meshed may be represented by an analytic function (parametric tensor-product spline surfaces) (Farin, 1996) or by discrete data, i.e. a triangulation (STL-data, subdivision surfaces). The resulting surface meshes are either directly used for simulation and analysis or serve as intermediate step for the generation of volume grids. However, the high quality criteria required for simulation (e.g. smooth variation of element size and shape all over the domain) are in most cases not satisﬁed by the direct outcome of surface mesh generation algorithms such as advancing front or Delaunay type methods. A common approach to provide the high mesh quality required for simulation is to perform surface enhancements which are therefore an essential prerequisite for the generation of surface meshes. Among the diﬀerent kinds of surface enhancements, smoothing is one of the mostly applied techniques.

Regarding two-dimensional domains, a large variety of smoothing techniques is available. An overview is given by Canann et al. (1998) and George and Borouchaki (1995). For a three-dimensional domain the generation of surface meshes is more complex. In the case of an analytic surface description, a classiﬁcation into two major groups is possible, parameter space and direct three-dimensional mesh generation.

The parameter space approach takes advantage of the underlying uv presen- tation of an analytic surface. The mesh is thus computed within this two- dimensional parameter space using conventional smoothing techniques for two-dimensional space. The generation of isotropic elements is ensured by arc-length parameterizations (Farouki, 1997) or special metrics derived from the ﬁrst fundamental form of the surface enabling the measurement of lengths, angles and area of faces (Chen and Bishop, 1997; Cuilliere, 1998; George and Borouchaki, 1995; Tristano et al., 1998).

Direct three-dimensional approaches on the other hand generate surface elements directly on a pre-discretized geometry without taking into account the two-dimensional parameter space representation of the surface (Cass et al., 1996; Lau and Lo, 1996; Lau et al., 1997). Those algorithms may produce better results in case of highly distorted mappings between parametric and physical space.

In both cases the analytic surface description facilitates smoothing as the surface description may be consulted for the placement of nodes and thus guarantees that the shape of an object is retained.

In contrary, when dealing with discrete surfaces no analytic surface description is available which could be utilized for smoothing. Up to now indirect mesh generation approaches were thus restricted to two dimensions.

The aim of this paper therefore is to provide a smoothing approach applicable for three-dimensional discrete surfaces to enable indirect mesh generation for arbitrary three-dimensional domains. This is achieved by locally approximat-

2 ing an analytic surface in the vicinity of the node intended for smoothing. Smoothing is then performed by moving the node on this approximated surface. However, the approach for local surface approximation presented in this paper is not only suitable for three-dimensional smoothing which is chieﬂy investigated in this paper but also for related problems such as point placement during mesh adaption or regridding of triangulations.

The article is organized as follows: In Section 2 the approach for local surface approximation is presented. The combination of established smoothing techniques with this approach is investigated in Section 3. The techniques developed in this context were integrated into an indirect quadrilateral mesh generator to demonstrate their robustness and practical relevance. The results derived from this approach are presented in Section 4.

2 Local Surface Approximation

The input for indirect quadrilateral mesh generation are triangular meshes. An essential prerequisite for successfully generating a quadrilateral mesh from the given triangular background mesh are adequate smoothing techniques. Exist- ing smoothing techniques for the indirect generation of quadrilateral surface meshes are only applicable for two-dimensional domains. For this reason a novel approach for three-dimensional smoothing is presented providing the possibility to process arbitrary triangular meshes in three-dimensional space. In order to smooth a node Ns in three-dimensional space, a local surface approximation is computed. Smoothing is then performed by repositioning node Ns on the approximated surface. The steps that are necessary to locally approximate a surface for smoothing of a node are described in the current Section.

2.1 Taylor Polynomial

The basic idea of three-dimensional smoothing is to move the node that has to be smoothed on an analytic surface that locally approximates the geometry. For each node Ns that has to be smoothed an analytic surface has to be determined that contains the node itself and that best ﬁts all direct neighbor nodes Ni connected to node Ns by an edge (Figure 1). A biquadratic Taylor polynomial is used to locally approximate the analytic surface:

u2 v2 F (u, v) = uF + vF + F + uvF + F (1) u v 2 uu uv 2 vv

3 The two-dimensional parameter space coordinates of a node are denoted by u, v. The computation of the two-dimensional parameter space coordinates of the center node Ns and its neighbor nodes Ni is described in Section 2.2.

The polynomial coeﬃcients Fu, Fv, Fuu, Fuv and Fvv are the ﬁrst and second order partial derivatives of F (u, v). A method to obtain these derivatives is presented in Section 2.3.

N5 NS N1

N4 N2 N3

Fig. 1. Three-dimensional smoothing scenario. The node selected for smoothing is denoted with Ns. Smoothing is performed by repositioning Ns on an analytic surface that runs through Ns and best ﬁts its neighbor nodes Ni.

2.2 Parameterization

For the computation of an approximated surface represented by the biquadratic Taylor polynomial given in equation (1), an almost isometric parameterization F (ui, vi) = Ni of the neighborhood of Ns has to be determined. A parameterization is thereby called almost isometric if kFuk ≈ 1, kFvk ≈ 1 and FuFv ≈ 0. For all further computations a local coordinate system is utilized with the property that the center node Ns is positioned at the origin in three-dimensional space as well as in two-dimensional parameter space, i.e. Ns = (0, 0, 0) = F (0, 0) = F (us, vs). As regards the choice of a parameterization there are several possibilities. In the following sections two alternatives are discussed, a projection into the tangential plane of Ns (2.2.1) and an exponential projection (2.2.2).

2.2.1 Projection into Tangential Plane

The most evident way to compute a parameterization is to project all neighbor nodes Ni into the tangential plane of Ns. The normal vector ~n of this tangential plane is derived by averaging the normals of the faces around Ns. A parameterization is thus given by the transformation of the projected nodes into an orthonormal basis with ~n being one of the basis vectors. The selection

4 of an orthonormal basis thereby ensures that the resulting parameterization is almost isometric.

A drawback of this method is that the projection into the tangential plane does not necessarily maintain the order of the neighbor nodes Ni around Ns. This might occur if the mesh is not ﬂat enough or if an improper tangential plane was chosen. Figure 2 shows such a situation.

0 4 5 1 2 3

5 0 2 3 1

Fig. 2. Projection into tangential plane does not necessarily maintain the order of neighbor nodes Ni around Ns.

2.2.2 Exponential Projection

A second parameterization that takes the lengths and angles between neighbor edges into account is the exponential projection. In contrary to the projection into the tangential plane the order of neighbor nodes Ni around Ns = (0, 0, 0) is maintained. For every neighbor node Ni the coordinates (ui, vi) within two- dimensional parameter space are computed as follows:

i 1 i 1 − − exp(N ) 7→ kN k cos( α˜ ), sin( α˜ ) (2) i i  j j  j=1 j=1 X X  

The angles α˜i are thereby computed by applying a function flat with

α˜i = ﬂat(∠(Ni, Ni+1)) = ﬂat(αi) (3)

where ∠(Ni, Ni+1) is the angle between Ni and Ni+1. The function flat scales the angles αi between two neighbor nodes Ni and Ni+1 so that the sum of the resulting angles α˜i amounts to 2π (Figure 3). In this way, a parameterization F (exp(Ni)) = Ni is deﬁned. Do Carmo (1976) showed that the exponential projection of equation 2 is an almost isometric parameterization.

5 As regards the choice of flat, a rather obvious possibility is given by uniform scaling: 2π α˜i = flat(αi) = αi (4) j αj P This definition of flat scales the angles in a uniform manner so that the sum of angles α˜i amounts to 2π. This approach works for any configuration of neighbors around a node and maintains the order of neighbor nodes around the center node (Welch and Witkin, 1994).

α i flat

~ αi

Fig. 3. Function flat scales the angles αi between two neighbor nodes Ni and Ni+1. Resulting angles α˜i amount to 2π. Another possibility to deﬁne the transformation flat is given by

α˜i = ﬂat(αi) = αi − , = αi − 2π . (5) n ! Xi

The diﬀerence between the angles αi in three-dimensional space and the angles α˜i in two-dimensional space is computed and equally distributed among all n angles αi, 0 ≤ i ≤ n − 1. This method might be less straightfor- ward than the uniform scaling of equation (4). Nevertheless, from a certain point of view this deﬁnition of the function flat shows an desirable behav- T n ior: flat is a projection of the vector (α0, . . . , αn 1) = (αi)i ∈ R+ con- − taining all n angles αi within three-dimensional space into the hyperplane Rn 1 2π H := { x ∈ + | hx|( √n )ii − √n = 0 }. In this hyperplane the sum of angles amounts to 2π, i.e. x ∈ H ⇔ kxk = 2π respectively. A formal proof of these considerations is given by Roessl (1999).

The last definition of flat thus determines the approximated angles α˜i in two-dimensional space having the smallest average quadratic error compared to the original angles αi in three-dimensional space. However, in the case of highly asymmetric configurations with angles αi < this definition of flat generates negative angles α˜i = flat(αi) < 0. In such a situation the order of nodes Ni around Ns is - contrary to the definition of flat in equation (4)

6 - not necessarily maintained. For this reason the deﬁnition of function flat presented in equation (4) was ﬁnally considered to be the better choice and was therefore preferred to the latter alternative.

2.3 Computation of Derivatives

After having determined the parameterization by computing the exponential projection described above, the derivatives Fu, Fv, Fuu, Fuv and Fvv are determined by solving the linear system of equations

VF~ = N . (6)

The rows of matrix V contain for each neighbor node i the coeﬃcients 2 2 ui vi (ui, vi, 2 , uivi, 2 ). The matrix N contains the 3D coordinates Ni of all neighbor nodes. After solving the system of equations the vector F~ contains the T derivatives, i.e. F~ = (Fu, Fv, Fuu, Fuv, Fvv) . The solution of the system of equations is computed using a Least Squares or Least Norm approach

T T 1 V (VV )− N : n < 5  F~ =  1 (7)  V− N : n = 5   T 1 T (V V)− V N : n > 5    where n is the total number of neighbor nodes Ni, 0 ≤ i ≤ n−1. Alternatively, as suggested by Welch and Witkin (1994), a different set of basis functions, 1 2 2 e.g. V = (ui, vi, 2 (ui + vi ))i may be used if v is badly conditioned or if less than five neighbor nodes are available. The resulting vector F~ is the solution of the linear system of equations and contains the approximation of the Taylor series coefficients Fu, Fv, Fuu, Fuv and Fvv of the surface F at Ns.

Extensive tests with a number of diﬀerent neighbor node conﬁgurations showed that this method generates partial derivatives with {Fu, Fv} and {Fuu, Fuv, Fvv} which are almost orthogonal. This observation is consistent with previous ex- pectations as the considered parameterization is almost isometric.

The exponential projection described in Section 2.2.2 is a parameterization of the center node and its neighbors and allows to transform the coordinates of the center node and its neighbors to two-dimensional parameter space. The Taylor polynomial of equation (1) containing derivatives of ﬁrst and second order that are computed by a Least Squares approach may be used to transform the two-dimensional parameter space coordinates of Ns back to three-dimensional space.

7 3 Smoothing

Smoothing is an important processing step for every surface mesh generator. Hence it is not surprising that a signiﬁcant amount of work has been done in this area. A good overview of some of that work is given by Canann et al. (1994), Canann et al. (1998) or George and Borouchaki (1995). As regards the indirect generation of quadrilateral meshes the smoothing techniques presented by Blacker and Stephenson (1991) and Owen et al. (1999) are of special interest. A common characteristic of those smoothing techniques is that they were originally designed for two dimensions. However, by using the approach for local surface approximation presented in this paper it is possible to adapt these techniques to curved surfaces in three dimensions, i.e. to perform smoothing in three-dimensions. The problem arising with this approach is that the exponential projection that transforms a node and its neighbor nodes to two-dimensional space does not preserve angles or lengths. Whereas close to Ns an Euclidian space may be assumed (the exponential projection is almost isometric with kFuk ≈ 1, kFvk ≈ 1 and FuFv ≈ 0) the mapping gets more distorted with growing distance to the center node Ns. For this reason the two-dimensional smoothing techniques have to be thoroughly ana- lyzed and adapted to the local surface approximation to obtain the intended smoothing results. In the following sections several two-dimensional smoothing techniques are combined with the surface approximation approach presented in this paper. The three-dimensional smoothing techniques derived from this approach have been integrated into an algorithm for three-dimensional quadrilateral mesh generation. The results achieved with this approach are presented in Section 4.

3.1 Laplacian Smoothing

Laplacian smoothing is by far the most common smoothing technique (Canann et al., 1998; Field, 1988; Freitag, 1997). The simplest form of Laplacian smoothing consists of placing the node Ns to be smoothed at the average of the n nodes connected to it. The movement of the center node Ns is thus described by a vector denoted with V~res resulting from the following equation:

n 1 − V~res = ( V~i) / n (8) i=0 X

This smoothing technique is usually applied to nodes that are surrounded by only one type of surface elements, triangles or quadrilaterals. The principle of Laplacian smoothing is shown in Figure 4.

8 N3

N2 V2 Vres N0 NS V0

Fig. 4. Laplacian smoothing. The node selected for smoothing Ns is repositioned at the average of its neighbor nodes connected to Ns by an edge. Three-dimensional Laplacian smoothing in combination with the surface approximation presented in this paper is performed by ﬁrst transforming all participating nodes into two-dimensional parameter space using the exponential projection of equation (2). Two-dimensional Laplacian smoothing is then performed within two-dimensional parameter space. The smoothed (u, v) coordinates of node Ns are then transferred back to three dimensions by computing the Taylor polynomial of equation (1).

3.2 Length Smoothing

Length smoothing is a technique that is applied if the node selected for smoothing has two adjacent quadrilaterals. The aim of this smoothing technique is to scale the edge between two adjacent quadrilaterals to a length l that suits the current edge length within the mesh. The two-dimensional approach proposed by Owen et al. (1999) takes into account the length of edges in the vicinity of Ns (Figure 5). Thereby, the average edge length of the surrounding edges (marked blue in Figure 5) is computed. The average length is then used to scale the selected edge (marked red in Figure 5) between the two adjacent quadrilaterals which is accomplished by repositioning node Ns. This approach is described by the following equation:

n 1 kV~ k + kV~ +1k + kV~ +2k + kV~ +3k + − kV~ k l = n n n n i=0 i (9) 4 + n P

In order to adapt length smoothing to the three dimensional case, it has to be taken into account that the exponential projection does not preserve lengths exactly. To solve this problem, the scaling factor by which the selected edge has to be stretched is computed in three-dimensional space. The scaling factor is then applied to the edge in two-dimensional parameter space which results in an adjustment of the coordinates (us, vs) of node Ns. This smoothed position is then transformed back to three dimensions by using the Taylor polynomial of equation (1).

9 V0 Vn−1

Ns Vn+3 Vn

Vn+2 Vn+1

Fig. 5. Length smoothing. The marked edge (red) is scaled to the average length of the surrounding edges (blue) by repositioning node Ns.

3.3 Angle Smoothing

Angle smoothing improves the angles of quadrilaterals and was ﬁrst introduced by Blacker and Stephenson (1991). This smoothing technique is performed if the node Ns selected for smoothing has two adjacent quadrilaterals. The processing steps for classical two-dimensional angular smoothing are outlined in the upper part of Figure 6. In a ﬁrst step, both vectors V~0 and V~1 are normalized and averaged. The resulting vector V~i denotes the ideal direction of the intermediate edge of the two adjacent quadrilaterals. Angle smoothing repo- ~ sitions node Ns at the tip of vector Vs0 which again is the average direction between the real direction V~s of the intermediate edge and the optimal direc- ~ ~ ~ ~ tion Vi. For the computation of Vs0 the vectors Vi and Vs are normalized and ~ ~ averaged. Vs0 is then scaled to the same length as Vs and Ns is moved to its ~ smoothed position at the tip of Vs0.

V's

V V Vs 0 i V1

V's

V Vi s

V0 V1

Fig. 6. Angle smoothing. Above: Original two-dimensional approach for angle smoothing. Below: Modiﬁed version for three-dimensional angle smoothing in combination with surface approximation.

10 In combination with the surface approximation the original approach for two- dimensional angle smoothing only yields moderate smoothing results. The reason for this behavior is that the transformation to two-dimensional parameter space does not preserve angles. To cope with this problem, a diﬀerent set of vectors V~0 and V~1 was selected for the computation of the optimal edge direction V~i. The quadrilateral diagonals V~0 and V~1 were replaced with the bot- tom lines of the respective quadrilaterals as shown in the lower part Figure 6, which proved to produce better smoothing results. This modiﬁed smoothing technique is applied to the two-dimensional parameter space coordinates of Ns.

3.4 Isoparametric Smoothing

The aim of isoparametric smoothing is to adjust nodes such that the quadrilateral surface elements are closer to parallelepipeds. The two-dimensional approach of isoparametric smoothing (Figure 7) was ﬁrst presented by Blacker and Stephenson (1991). The node intended for smoothing is denoted by Ns. The aim is to reposition Ns in such a way that opposite quadrilateral edges are parallel after smoothing. If Ns only has one adjacent quadrilateral the smoothed position Ns0 of Ns is computed as follows:

~ ~ ~ Ns0 = Ns + (V0 + V1 − Vd) (10)

Otherwise, a generalization of this equation is used which is applicable to nodes Ns surrounded by n quadrilaterals:

n 1 1 − ~ ~ ~ Ns0 = Ns + V0i + V1i − Vdi (11) n i=0 ! X

Isoparametric smoothing is adapted to three dimensions by transferring all participating nodes to the two-dimensional parameter space using the exponential projection described in Section 2.2.2. In the case of isoparametric smoothing valuable smoothing results are achieved by performing smoothing in two-dimensional parameter space as described above and transforming the smoothed position back to three dimensions. Even though this smoothing technique does not create perfect parallelepipeds in three-dimensional space, in most cases a major improvement of element shape is achieved.

11 Ns V0

V1 Vd

Ns V0

V1 −Vd

Fig. 7. Isoparametric smoothing. This smoothing technique tries to position the node Ns selected for smoothing in such a way that the quadrilateral adjacent to Ns is a parallelepiped. 4 Results

In order to demonstrate the strengths of the techniques presented in this paper an indirect quadrilateral mesh generator was developed based on the approach presented by Owen et al. (1999). Our advancing-front mesh generator is able to convert triangular meshes in two-dimensional and three-dimensional space into quadrilateral meshes. Smoothing operations are performed after the creation of each new quadrilateral. Depending on the properties of the mesh in the vicinity of the node selected for smoothing the mesh generator selects an appropriate smoothing technique. In most cases the mesh generator combines diﬀerent smoothing techniques to determine the new position of the node.

The algorithm was extensively tested with a large number of triangular input meshes in three-dimensional space. Each one of these meshes showed diﬀer- ent geometric and topological features such as holes, strong curvature, sharp angles, etc. A selection of three-dimensional quadrilateral meshes generated with our algorithm is shown in Figures 8 and 9.

To assess the quality of three-dimensional smoothing the forward Hausdorff distance between the initial triangular mesh and the quadrilateral mesh which for this purpose was decomposed into triangles (each quadrilateral was divided into two triangles) was computed. The measurement of Hausdorff distances was thereby performed using M.E.S.H. (Aspert et al., 2002). The resulting Hausdorff distances for the two example meshes in Figure 8 and 9 are given in Table 1. In the table the forward Hausdorff distance is compared to the

12 size of the bounding box diagonal enclosing the surface mesh. The error for 3 the deltawing example is approximately 1000 of the bounding box diagonal, 2 for the other model about 100 respectively. This clearly shows that the error introduced by the local surface approximation approach is negligible.

Hausdorﬀ distance Air Plane Deltawing

Absolute 0.00674602 0.00128552 % BBox diag 0.0172316 0.00262416 Table 1 Hausdorﬀ distances computed for the two example meshes in Figure 8 and 9. These examples demonstrate that the approach for three-dimensional smoothing is robust and preserves the shape of the surface geometry. The analytic surface approximation presented in this paper is a useful tool which in combination with known and well established 2D smoothing techniques allows for the indirect generation of surface meshes in three dimensions.

5 Conclusions

In this paper a new approach for three-dimensional smoothing is presented. An analytic surface is computed that locally approximates the geometry of the mesh in the vicinity of the node selected for smoothing. Smoothing is then performed by repositioning the concerning node on the approximated surface. Conventional two-dimensional smoothing techniques in combination with the local surface approximation proved to be well suited for node smoothing in 3D. In some cases slight modiﬁcations of the respective smoothing technique were necessary to improve smoothing results. The quality of the resulting surface mesh was assessed by measuring the Hausdorﬀ distance between the initial triangular mesh and the output mesh. For the indirect generation of surface meshes the approach presented in this paper is of high value, especially if only legacy three-dimensional triangular meshes are available.

In addition, the presented approach for local surface approximation could also be beneﬁcial for related problems such as point placement during mesh adaption or regridding of triangulations which will be subject of future research.

6 Acknowledgments

This work was partly supported by the Deutsche Forschungsgemeinschaft (Graduate Research Center 244) and EADS Military Aircraft, Munich.

13 References

Aspert, N., Santa-Cruz, D., Ebrahimi, T., MESH: Measuring Error between Surfaces using the Hausdorff distance, Proc. IEEE Multimedia and Expo (ICME), 1, 705-708, 2002. Blacker, T.D., Stephenson, M.B., Paving: A new Approach to Automated Quadrilateral Mesh Generation, Int. J. Numer. Meth. Eng., 32, 811-847, 1991. Canann, S.A., Muthukrishnan, S.N., Phillips, R., Topological improvement procedures for quadrilateral and triangular finite element meshes, Proc. 3rd Int. Meshing Roundtable, 1994. Canann, S.A., Tristano, J.R., Staten, M.L., An Approach to Combined Lapla- cian and Optimization-Based Smoothing for Triangular, Quadrilateral, and Quad-Dominant Meshes, Proc. 7th Int. Meshing Roundtable, 1998. Cass, R.J., Benzley, S.E., Meyers, R.J., Blacker, T.D., Generalized 3-D Paving: An Automated Quadrilateral Mesh Generation Algorithm, Int. J. Num. Meth. Eng., 39, 1475-1489, 1996. Chen, H., Bishop, J., Delaunay Triangulation for Curved Surfaces, Proc. 6th Int. Meshing Roundtable, 1997. Cuilliere, J.C., An adaptive method for the automatic triangulation of 3D parametric surfaces, Computer-Aided Design, Elsevier, 30, 2, 139-149, 1998. Do Carmo, M.P., Differential Geometry of Curves and Surfaces, Prentice Hall, 1976. Farin, G.E., Curves and Surfaces in Computer Aided Geometric Design, Aca- demic Press Inc., 1996. Farouki, R.T., Optimal parameterizations, Computer Aided Geometric De- sign, 14, 2, 153-168, 1997. Field, D., Laplacian smoothing and Delaunay triangulations, Communications in Num. Meth. in Eng., 4, 709-712, 1988. Freitag, L., On Combining Laplacian and Optimization-Based Mesh Smooth- ing Techniques, AMD Trends in Unstructured Mesh Generation, ASME, 220, 37-43, 1997. George, P.L., Borouchaki, H., Delaunay Triangulation and Meshing, Applica- tion to Finite Elements, Editions Hermes, Paris, France, 230-234, 1995. Lau, T.S., Lo, S.H., Finite Element Mesh Generation Over Analytical Surfaces, Computers and Structures, Elsevier, 59, 2, 301-309, 1996. Lau, T.S., Lo, S.H., Lee, C.K., Generation of Quadrilateral Mesh over Ana- lytical Curved Surfaces, Finite Elements in Analysis and Design, Elsevier, 27, 251-272, 1997. Owen, S.J., Staten, M.L., Canann, S.A., Saigal, S., Q-MORPH: an indirect approach to advancing front quad meshing, Int. J. Numer. Meth. Eng., 44, 1317-1340, 1999. Roessl, C., Semi-Automatische Methoden fuer die Rekonstruktion von CAD- Modellen aus Punktdaten, Diploma Thesis, University of Erlangen, 1999. Tristano, J.R., Owen, S.J., Canann, S.A., Advancing Front Surface Mesh Gen-

14 eration in Parametric Space Using a Riemannian Surface Deﬁnition, Proc. 7th Int. Meshing Roundtable, 1998. Welch, W., Witkin, A., Free-Form Shape Design Using Triangulated Surfaces, Computer Graphics, Annual Conference Series, 28, 247-256, 1994.

15 Fig. 8. Air plane example. The left column shows diﬀerent views of the triangular background mesh. On the right the respective view after indirect quad mesh generation using the 3D smoothing approach introduced in this paper is shown.

16 Fig. 9. Deltawing example. The left column shows diﬀerent views of the triangular background mesh. On the right the respective view after indirect quad mesh generation using the 3D smoothing approach introduced in this paper is shown.