Modified Catmull-Clark Methods for Modelling, Reparameterization and Grid Generation

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1 Modified Catmull-Clark Methods for Modelling, Reparameterization and Grid Generation Karl-Heinz Brakhage RWTH Aachen, 55 Aachen, Deutschland, Abstract In this paper we demonstrate that subdivision schemes can be used for modelling and grid generation in aeronautical projects such as that in the collaborative research center SFB at the RWTH Aachen: F low Modulation and Fluid-Structure Interaction at Airplane Wings. To make the schemes usable for this purposes their rules have to be modified in such a way that the limit surfaces have the desired properties. Furthermore fast algorithms for evaluation and approximation of existing surfaces have to be developed. Introduction We start with a short survey on the history of subdivision. At least fifty years ago the first papers on subdivision were published. In 97 de Rham s first articles on corner cutting appeared []. He suggested to do corner cutting with ratio : : compare Figure ). From the start polygon with vertices P, P,..., P n all the vertices P i are split the following way the indices of the old vertices P i have to be computed modulo n) Qn = P P + P n )/ Q = P + P )/ P P i.. Q = P + P )/ Q = P + P )/ Qi = P i + P i )/ Q i = P i + P i+ )/ ) Since the type of the refinement rule for this task is a vertex split we call it a vertex split scheme. The limit curve of this process is only GC- this is the same as C-, the curve is continuous). Later de Rham generalized the scheme by introducing a weight ω and uses the ratios ω : ω : ω. Furthermore he showed that the limit curves of this corner cutting are GC- this is C- continuously differentiable) after reparameterization) if and only if ω = /. The relevance of such algorithms to curve generation became clear with the paper of the graphics artist G. Chaikin [] in 97. At the 97 CAGD conference at the University of Utah, he invented an algorithm for high speed curve generation which coincides with the de Rham method for ω = /. Already at that conference R. Riesenfeld and M. Sabin opined that Chaikins algorithm is an iterative way to generate uniform quadratic B-spline curves. An analysis of the method had the result that the limit curve is a uniform B-spline curve whose control points are the vertices of the starting polygon. Further investigations showed that uniform B-spline curves of any degree would have such a subdivision construction. This is shown for the cubic P Q Q R R R R n- Q n- R R n- R R 5 P n- Q n- Q 9 Q P Q Q 7 Q 5 P P Q Q Q P P n- P Q n- R n- R n- Q n- Figure : Cubic edge splitting scheme two steps). In every step all edges are split into two parts. The new points on the edges are constructed by ratio : of the adjacent vertices. The old vertices are recomputed with the ratio : :. Q R R Q P R R R R 5 Q Q Figure : Corner cutting of de Rham 97: The outer polygon P, P,..., P n ) is subdivided two times with the edge ratio ::. The result is the inner curve polygon R, R,..., R n ). We use the sequence P, Q, R,... instead of P, P, P,... to reduce the number of indices. Afterwards the outer parts corners) are cut away. The resulting polygon is the input for the next same step. P Q Q 5 P case in Figure. In this method the edges are split into two pieces and hence it is called edge split scheme. We have new vertices on the edges and recomputed old ones. The formulas for this scheme are Q = P n + P + P )/ Q = P + P )/. Q i = P i + P i + P i+ )/ Q i+ = P i + P i+ )/ ) The first subdivision schemes for surfaces were introduced in 97. Doo and Sabin [] established a quadratic

2 subdivision scheme which is a direct extension of the vertex split method of de Rham / Chaikin. Furthermore they analyzed the behavior of such schemes. This was done by writing the subdivision process in form of matrix multiplications and studying the eigenvalues and eigenvectors of the subdivision matrices. face point edge point WN)= NN-) Figure : Subdivision rules for the Catmull-Clark Method valence N = 5). The new) face points are the barycenters of the vertices of each face. This is denoted by averaging with the weights for every face vertex. Next the edge points are computed using the new face points according to the given weights. In the last vertex step the old) vertices are recomputed. Again the weights are given in the figure. Notice that in every case we have to divide by the sum of the weights. Catmull and Clark [] published their descriptions of quadratic and cubic subdivision surfaces in 97 too. Both papers appeared in the same journal on consequent pages.) The latter one regards to the curves generated according to ). Contrary to tensor-product splines their scheme can be applied to meshes that are not regular rectangular grids see Figure ). The number of edges that meet at a vertex is called the valence of that point. We denote it with N. After the vertex steps face vertex, edge vertex and vertex vertex) illustrated in Figure the new edges are build by splitting the old ones edge edge) and connecting the new face points with the new edge points face edge). The new faces are the faces inside the old ones face face). The right plot of Figure can be thought of as an example for a pentagon. The centroid is the new face vertex, the vertices with weight are the new edge vertices and the the vertices with weight are the recomputed vertices. In a regular rectangular grid the valence of all interior vertices is N =. Subdivision schemes that have the same coefficients/rules in every level are called stationary. In our modelling and grid generation concepts see [5]) we are interested in ending up with smooth untrimmed B-spline patches. Thus for our purposes the Catmull- Clark scheme is the method of choice. We summarize its crucial properties: The surfaces can be of arbitrary genus since the subdivision rules can be carried out on a mesh of arbitrary topological type. After one subdivision step all faces are quadrilaterals. Except at extraordinary vertices vertices of valence N ) the limiting surface converges to uniform bi-cubic B-Spline patches. Hence the surface is C except at extraordinary vertices. The number of extraordinary vertices is fixed after the first subdivision step. After two subdivision steps all faces have at most one extraordinary vertex. Near an extraordinary vertex the surface can be shown to have a well defined tangent plane at the limit point, but the curvature there is generally not well defined. The subdivision rules can be modified in such a way that they generate infinitely sharp creases as well as semi-sharp creases, i.e. creases whose sharpness can vary from zero meaning smooth) to infinite. See de Rose et. all [] 99) for more details. Because at least after one subdivision step there are only quadrilaterals we show the rules for this case in Figure. The correspondence to cubic B-splines can easily be seen. Notice that for the regular case A) =. AN) Figure : Subdivision rules for the Catmull-Clark-Method for quadrilaterals only. The weight for the recomputed extraordinary vertex of valence N is AN) = N 7 N. Again we have to divide by the sum of the weights in every case. Analysis of subdivision schemes The ideas of Doo and Sabin [] using eigenanalysis of the subdivision matrix were advanced by Ball and Storry from 9 to 9 in [7], [] and [9]) and later by Stam [] in 99. Stam additionaly gave an algorithm for evaluating the Catmull-Clark-Scheme and its derivatives) at arbitrary points. He used a choice of ordering for the control vertices that the main part of the subdivision matrix has a cyclical structure. Hence the discrete Fourier transform can be used to compute its eigenstructure. For our practical examples we can restrict the valence N of the extraordinary vertices to N, 5, }. For these cases we use a more intuitive numbering and pre-compute the eigenstructure with numerical methods. Our aim is Table : Dimension of subdivision matrices at extra ordinary points valence rows columns regular N N N + + 7

3 to solve interpolation and approximation problems for this methods. Thus we do not only need values at certain points but also need the coefficients of the involved mesh points. For this reason we additionally build up tables for the interpolation and approximation schemes for valence N,, 5, }. More details can be found in Appendix A CD-version of this paper only). For surfaces and even for valences N the subdivision matrices are of regarding analysis high dimension see Table ). Therefore we explain the ideas for surfaces and give the analysis for curves. The main difference between curves and surfaces is the existence of extraordinary points. The polygon for the curve subdivision can be evaluated everywhere at each step by well known B- spline algorithms whereas the surface mesh can not. The control vertex structure near an extraordinary point is not a simple rectangular grid, thus all faces that contain extraordinary vertices cannot be evaluated as uniform B-splines. Since after one subdivision step all faces are quadrilaterals and after two steps all faces have at most one extraordinary vertex, we assume for our mesh that each face is a quadrilateral and contains at most one extraordinary vertex. Figure 5 shows that the region Figure 5: Behavior of the Catmull-Clark-Method near an extraordinary point of valence N =. The non regular region dark) is scaled down with every step. where the surface can not be evaluated with standard methods is reduced in every subdivision step. Since we can evaluate the surface at extraordinary vertices the remaining problem is to demonstrate only how to evaluate a patch corresponding to a face with just one extraordinary vertex, such as the dark region shown in Figure 5. Analogue to curves compare Figure ) we introduce parameter values and define a surface patch xu, v) over the unit square [, ] [, ] such that the point x, ) corresponds to the extraordinary vertex. We can evaluate the surface at those vertices x, )) as a linear combination of the circumfluent vertices the same ones we need for the re-computation step vertex vertex during subdivision, but with different weights). Additionally we can evaluate xu, ) for u [, ] and x, v) for v [, ] as regular B-spline part. The remaining problem is the evaluation xu, v) in the rest of the unit square. This problem is solved in the following way. First we do just enough subdivision steps such that u, v) corresponds to a regular part at that stage. We then pick the right vertices and do the evaluation as regular B-spline. Q P P Q C Q Q C Figure : Cubic edge splitting scheme one step and limit curve xt), t [, ]). The points P, P, P, P determine the curve between C = P + P + P )/ and C = P + P + P )/. We say that C = x) corresponds to P and C = x) to P After one subdivision step C = x.5) can be computed as C = Q + Q + Q )/ = P + P + P + P )/. We will give the analysis for curves now see Figure ). Imagining P as an extraordinary vertex we can not compute xt), t [, ] by standard B-Spline methods cause we have to cross P and for this thought experiment we have an analogon to the surface case. The subdivision matrix A corresponding to P is given by A = and Q Q Q C P Q P = A P P P. ) The eigenvalues of A are, / and /. For every of our subdivision schemes there is a unique largest eigenvalue, all eigenvalues are real and the matrices are diagonalizable equal algebraic and geometric multiplicities for all eigenvalues). One matrix that has as columns the eigenvectors of A is V = V and thus = ) V A V = diag,, } = Λ. 5) On the other hand this yields A = V Λ V. Furthermore we have A n = V Λ V ) n = V Λ n V and lim n Λ n = diag,, } =: Λ. Now we can conclude A := lim n An = V Λ V =. ) If we apply the subdivision for the curve case on P, P, P this yields C = x) = P + P + P 7)

4 and we have proven the consistency to uniform B-spline curves. Due to the fact that λ = > λ = / > λ = / we can already conclude from the given inverse in ) that x) = P + P + P x ) = α P + P ) x ) = α P + P P ) ) If we want to do the same computation for C = x.5) we have to enlarge the subdivision matrix A to A with ) A A = = A A. 9) The new rows directly follow from the usual B-spline knot-insertion rules for curves and surfaces. We make use of the block structure to compute the eigenvalues and eigenvectors of A. First it induces ) ) V Λ V = and Λ = ) V V Λ as structure for the equation A V = ΛV. Inserting ) and expanding the last equation yields ) ) A V V Λ = A V + A V A V V Λ V Λ and we can compute the remaining eigenvalues and the eigenvector block V with A. For surfaces A is a 7 7 matrix and independent of the valence N. Hence it can be computed in advance and used for any valence. The remaining part is the extension V of the eigenvectors in V. V is a 7 N + matrix and can be computed by solving the N + linear systems A V + A V = V Λ ) which in more detail is A Λ j) I 7 ) V j) = A V. In the last equation we have used Λ j) for the j th diagonal element of Λ, I 7 for 7 7 identity matrix and V j) for the j th column of V. In our thought experiment for curves see 9)) we have A = /) = Λ and V = ). Solving equation ) for V, which is ) + V = V, we get V = ) and finally V =. ) For the computation of the inverse of V we can make use of the block structure again. We omit the details and only state the result V =. ) The same computations as above lead to V A V = diag,,, } = Λ. ) and A n = V Λ V ) n = V Λ n V. 5) Repeating the above process we can compute x.5) etc. But if we want to compute values in.5, ) we have to compute Q and thus need another subdivision step given by an usual B-spline knot-insertion rule. Notice that we do not need further vertices as input. Our matrix for such a step is A Â = A A = A A. ) Let us summarize the results from above. The first steps of the subdivision processes in matrix form are Q = A P and ˆQ = Â P. 7) We have collected the vertices in P old level) and in Q and ˆQ new level). If we build the product of the mask, /, /, /) and A we get the coefficient vector c = /, /, /, /) for P, the vector of vertices compare Figure ). The same result can be obtained by evaluating the basis functions of 9) for s = /. Since A is a square matrix we can iterate with it. Thus if we want to compute values inside t [/ i+, / i ] we have to do i subdivision steps with A and than one with Â. This results in a general subdivision matrix S = Â Ai = Â V Λi V. ) After that we pick the points numbered from to see Figure ) to compute the desired value by normal B- spline evaluation. The same way we can compute coefficients for the the vertices of the starting polyhedron everywhere cause we know the basis functions for cubic uniform B-splines. On [, ] they are b s) = s), b s) = + s) + s) s, b s) = + s + s s), b s) = s. 9)

5 We only have to transform the value t [/ i+, / i ] to s [, ] by the formula st) = t / i+ ) i+. In the event of surfaces we make use of the tensor product structure. More details can be found in Appendix A CDversion of this paper only). We now describe how to build the matrices for the approximation of given surfaces with our Catmull-Clark meshes. We can compute the values L i of the limit surface and the corresponding coefficients c j of the involved vertices for arbitrary wanted points. L i can be written as L i = c T i Pi where we have collected the c j in the vector c i and the associated vertices in P i. For a given surface s we can project L i onto it: L i L s i. Know we have the desired equation of form c T i P i = L s i. ) Sampling enough of these collocation points we end up in a sparse linear system for interpolation or an overdetermined sparse linear) system for approximation. Normally we use the latter one and solve it with CGLS, a Conjugate Gradient method for linear Least Squares also called CGNR in []). To apply CGLS for solving the normal equations we only need effective methods to multiply the system matrix M and its transpose M T with vectors. It should be noted that our vectors are vectors of d-vectors. In ) we can see how the sparse multiplication with M is done for one row. From Table we know that for valence N we have at most entries in each row of M. This number is independent of the number of control vertices #V). For the multiplication with M T we have to construct coefficient vectors for each row of M to get an analogue form to ). Again the number of non-zero entries is independent of #V but it depends on the ratio number of sample points)/#v. Figure : Simplified wing tip: Start polyhedron for the Catmull-Clark-Method and polyhedron after three subdivision steps =N+7 = N+ =N 5 II 9 I IV III 9 =N+ 7 =N+ 7=N+ 5 = N+ Figure 7: Area of effect for the Catmull-Clark method valence N =. For the computation of the limit point corresponding to vertex we need the N + = 7 vertices numbered from to N =. For area I we need the additional 7 this number is independent of N) vertices from N + = 7 to N + 7 =. For the computation inside the areas II, III and IV we need another 9 vertices from N + = to N + =. For valence N > we have a vertex 7 which does not exist in the case N = where we have to identify it with vertex. Examples In this section we give some examples from our aeronautical project. All examples show the vertex meshes and Figure 9: Wing-fuselage constellation: Start polyhedron for the Catmull-Clark-Method and constellation after three subdivision steps not the limit surfaces. Furthermore we do not project the subdivision surface onto our given constellation because we want to demonstrate the nice smoothing properties of the subdivision surfaces on a larger scale. For real applications an approximation step is done at the state of the lower parts of the figures and for flow calculations an exact evaluation of the limit surface is done. Figure shows the reparameterization of a wing tip. Some details of Figure 9 can be seen, especially the high grid quality at the extraordinary points valence ) of the wing tip. Furthermore it can be seen that already a very rough approximation leads to a high quality grid. Figure 9 shows a part of a wing-fuselage configuration. We use a non stationary subdivision process with tagged edges, in other words: some edges are treated specially for a few steps. For the first steps for instance we treat

6 Aachen, University of Technology, Aachen, Germany. Figure : Approximation of an engine: Start polyhedron for the Catmull-Clark-Method and polyhedron after three subdivision steps the transition curve from the wing to the fuselage with the masks of ) as a curve. Only in the last step we use the surface masks. This leads to semi sharp part in that region. Tagged edges and their influence on the limit surface where analyzed in [], [] and []. We use a modified version of the latter on. Figure shows the approximation of an engine. The scalings of the different areas have been slightly modified to see some details, namely the sharp edges and the overall high grid quality. Conclusion We have presented the basic framework for modelling, interpolation, approximation and grid generation with Catmull-Clark methods in technical projects. The result is a watertight geometry of untrimmed surface patches. The isolines of them yield high quality block structured grids that are well suited for adaptive flow solvers. In this contents adaptation is simply function evaluation. Our approach is based on the precalculation of coefficient masks for standard or modified Catmull-Clark methods. These coefficients can directly be used to efficiently solve the sparse linear systems. Acknowledgments This work has been performed with funding by the Deutsche Forschungsgemeinschaft in the Collaborative Research Center SFB Flow Modulation and Fluid- Structure Interaction at Airplane Wings of the RWTH References [] G. de Rham. Un peu de mathématiques á propos d une courbe plane. Elemente der Mathematik, ):9, 97. [] G. Chaikin. An algorithm for high speed curve generation. Computer Graphics and Image Processing, : 9, 97. [] M.A. Sabin and D. Doo. Behaviour of recursive division surfaces near extraordinary points. CAD, ):5, 97. [] J. Clark and E. Catmull. Recursively generated B-spline surfaces on arbitrary topological meshes. CAD, ):5 55, 97. [5] K.-H. Brakhage and Ph. Lamby. Application of B-spline techniques to the modeling of airplane wings and numerical grid generation. submitted to CAGD, 7, preprint at: [] T. DeRose, M. Kass, and T. Truong. Subdivision surfaces in character animation. In M. Cohen, editor, Proceedings of the 5th annual conference on Computer graphics and interactive techniques, ACM SIGGRAPH, pages 5 9. Addison Wesley, 99. [7] A.A. Ball and D.J.T. Storry. Recursively generated B-spline surfaces. In Proceedings of CAD, pages 9, Brighton, Sussex, U.K., April [] D.J.T. Storry. B-spline surfaces over an irregular topology by recursive subdivision. Ph.d. thesis, Loughborough University, 9. [9] A.A. Ball and D.J.T. Storry. Conditions for tangent plane continuity over recursively generated B-spline surfaces. ACM Transactions on Graphics, 7):, 9. [] J. Stam. Exact evaluation of catmull-clark subdivision surfaces at arbitrary parameter values. In Proceedings of SIGGRAPH, pages 95, revised version on: [] Y. Saad. Iterative Methods for Sparse Linear Systems. SIAM, second edition,. [] H. Hoppe, T. DeRose, T. Duchamp, M. Halstead, H. Jin, J. McDonald, J. Schweitzer, and W. Stuetzle. Piecewise smooth surface reconsruction. In Computer Graphics Proceedings, Annual Conference Series, ACM SIGGRAPH, page 95. Addison Wesley, 99. [] H. Biermann, A. Levin, and D. Zorin. Piecewisesmooth subdivision surfaces with normal control. In Kurt Akeley, editor, Siggraph, Computer Graphics Proceedings, ACM SIGGRAPH, pages. Addison Wesley Longman,.