Constructing 3D Surfaces from Random Sets of Points by Means of Level Curves CAZACU Razvan a and GRAMA Lucian b
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1 Applied Mechanics and Materials Online: ISSN: , Vol. 371, pp doi: / Trans Tech Publications, Switzerland Constructing 3D Surfaces from Random Sets of Points by Means of Level Curves CAZACU Razvan a and GRAMA Lucian b Industrial Engineering and Management Department, Petru Maior University, Nicolae Iorga Street, No. 1, Targu Mures, Romania a razvan.cazacu@ing.upm.ro, b lucian.grama@ing.upm.ro Keywords: Delaunay triangulation, level curves, 3D surface, CAD, CMM, VB.NET. Abstract. This paper proposes an easy approach to obtaining 3D surface models in parametric CAD software products from random sets of points, based on the analogy of level curves from geodesic disciplines. The points could be obtained from measurements on existing mechanical parts, as a stage in reverse engineering processes, or can be defined arbitrary in space, as part of the design process. The present research shows that good approximations of the true surface can be obtained by measuring a relative small group of carefully selected points and constructing level curves, following the same procedure as in the case of topography. The instrument to build level curves is an original application developed for AutoCAD using VB.NET which allows the user to construct the level curves parallel to the xy plane for any number of points in space. We tested the application on more set of points sampling the same surface, obtaining the level curves in AutoCAD and using these to build the final 3D model. The surfaces obtained this way were then compared to draw some conclusions regarding the influence of the number of sampled points on the precision of the model. Introduction When measuring existing mechanical parts, the coordinate measuring machines (CMM) are typically used for small models, but in the case of large scale parts (on the order of meters) the alternative mobile spatial coordinate measuring system (MScMS) is more appropriate [1]. In both cases however, the number of points to be measured is very large and a proper measuring strategy is needed to ensure enough data points are obtained while the amount of saved data and measurement time are kept within decent limits [2, 3]. The choice of the right measuring machine can be done automatically, by the use of trained neural networks [4]. Following the analogy to topography and starting from the cluster of points imported in AutoCAD, we developed an application that constructs the Delaunay triangulation [5, 6] and based on this arbitrarily dense level curves for the model as cubic splines [7]. The application is written in full OOP paradigm using VB.NET and the managed AutoCAD libraries for.net, this being the newest, most powerful and flexible way of writing code for AutoCAD [8]. After the generation of the level curves in AutoCAD, these can be exported to any 3D parametric modeling software where they are used to build the final surface of the model. Getting the Points The cluster of points can be a result of measuring existing surfaces using CMMs, as a part of the reverse engineering process or can be defined in space as a part of the creation process. In both cases a good strategy for choosing the points to read or define can increase the precision of the final model. Following the same analogy to topography [9] (which has also been extended to other disciplines [10]), it s common to establish a main direction and take parallel sections along that direction. On each section, one takes several points, preferably at the spots where the section profile visibly changes its gradient. The density of points should be proportional with the curvature of the profile. On the zones with greater curvature, more points are needed for a good approximation; on the zones where the curvature is not so big, a smaller number of points are enough. This whole procedure is visually represented in Fig. 1. All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, (# , Pennsylvania State University, University Park, USA-13/09/16,08:06:07)
2 484 Innovative Manufacturing Engineering a) Taking sections b) Taking points along the sections Fig. 1. Strategy for sampling points from existing surfaces. More dense sections and points along sections mean more precise final representations, but a more tedious and long process at the same time. In order to illustrate the difference in precision, we will consider in this research two sets of points for the same surface, one with 200 points and one with 400 points. In order to be able to quickly and accurately sample points and calculate differences between the real surface and the approximated one, we considered an analytic surface as the model for our case study. The surface is given by the multiplication of two planar functions:,= (1) The two functions fx(x) and fy(y) are the result of the interpolation with the Lagrange polynomial of a number of discrete points, 5 along the x axis and 15 along the y axis. The constant C is for scaling the surface in accordance with the desired result. In order to take sample points along the surface, we wrote a program for AutoCAD using VB.NET. The program allows the user to set the function for the surface, the number of sections and the number of points along the sections and draws inside the AutoCAD file the respective points at the appropriate locations. These are then used by the next application to build the level curves. Building the Level Curves In order to build the level curves starting from the sampled points we developed an application using VB.NET, leveraging the power and flexibility of the.net framework and object oriented programming in general [11]. The application targets the AutoCAD environment and works inside a session of this software. AutoCAD offers access to its infrastructure in several ways and can be programmed using many different technologies (Lips, VBA,.NET, etc.). For programming inside.net, it offers two managed libraries (acmgd.dll and acdbmgd.dll) that expose its environment as managed classes. Other applications using different tools also exist [12]. Once the application is loaded, the command lc (defined by the application) can be invoked as any native ACAD command. The user is required to select the points in space and to specify how dense the level curves should be. Based on this data the algorithm calculates and draws the set of level curves as cubic splines, using AutoCAD s Spline entity. The algorithm works in 3 steps: Build the Delaunay triangulation of the cluster of points; Find the points along the triangulation lines who belong to the needed level curves; Connect the points on the same level curve using a cubic spline.
3 Applied Mechanics and Materials Vol The Delaunay Triangulation (DT). DT is used to connect sets of points in plane (or higher dimensions) by a network of triangles that maximize the minimum angle of all triangles in the triangulation. This property is crucial to obtain a good approximation of the real surface. Although the points are defined in space, we are looking only for their position in the xy plane, the z axis being used to build the intermediate points along the level curves. For a Delaunay triangulation, which has a reciprocal relation to the corresponding Voronoi tessellation (obtained by connecting the center of the circumcircles of all triangles), also has the property that no point lays inside the circumcircle of any of the triangles. There are many algorithms for building the Delaunay triangulation starting from a set of points [13, 14, 15]. The most common are: flip algorithms, incremental, divide and conquer (fastest), sweep line, etc. For this application we chose a sweep line algorithm, with the advantage to be both fairly simple to implement (simpler the divide and conquer, more complex than flip) and efficient (less than divide and conquer, more than flip). The result is saved in memory using a structure that makes it easy to manipulate the elements of the triangulation. The intermediate points. The points are calculated for each line in the triangulation using a linear interpolation. Considering the coordinates of the two connected points and which level curves will pass between them, the program calculates the x and y position of the points on each level curve and save them as part of the line properties. The final level curves. The curves are obtained by connecting the adjacent points on the same level curve by cubic splines. To find the points, the program uses the information from the structures saved in memory in the previous steps. The Spline entity is used to draw the closed level curves along the whole surface. Using the application for the 2 set of points obtained in the previous step (200 and 400 number of points), the level curves represented in Fig. 2 were obtained. a) 200 sampled points b) 400 sampled points Fig. 2. Obtained level curves in the 2 cases. Constructing the 3D Model After we have the level curves in AutoCAD, they can be imported in most CAD software programs (CATIA, ProEngineer, SolidWorks, SolidEdge, etc.), either directly as dwg files or through the intermediate dxf, exchange file type. All 3D modeling programs offer a command to build solids or surfaces from a set of sections. We used AutoCAD Loft command for this purpose. In the case the sections (level curves) are open, the software builds a surface; if they are closed, the resulting object is a solid. We obtained a solid as our level curves are closed. The resulting geometry is shown in Fig. 3, in a realistic presentation. It can be observed that the two solids, obtained for the 200 and 400 sets of points look similar. The differences are not visually obvious and will be analyzed in the next section.
4 486 Innovative Manufacturing Engineering a) 200 sampled points b) 400 sampled points Fig. 3. 3D model obtained from level curves Considerations About Precision With the increase of the number of sampled points (number of sections and number of points along one section) the precision of the final model also increases. On the other hand, the computational effort is also bigger and the advantage of this procedure is exactly the need for fewer points than with other methods. We calculated the maximum difference between the real model (the analytical original function) and the approximated model in the two cases. The graph drawn in Fig. 4 clearly shows that the maximum error (in millimeters) is lower in the case of fewer sampled points (case 2). Although the method is not suitable for high precision modeling, it can be successfully used for bigger models where the overall precision of the representation is not that much of an issue. The clear advantage is that it requires much fewer sampled points than in the classical cases of surface approximation. Conclusions Fig. 4. Influence of number of points on precision The present paper proposes an alternative method for building 3D models out of sampled points (obtained either from real models with the help of coordinate measuring machines, or defined directly in space). The method is based on the analogy with topography, where the surfaces are recreated by measuring the points along sections of the 3D model where the profile presents visible changes in the gradient. Those points are then processed with a software program which creates the level curves as a means of terrain representation. For mechanical engineering purposes, the level curves can be processed by 3D modeling programs to construct the approximate representation of the original model. The points considered in this research were obtained by sampling points along an analytical surface, defined by two planar and orthogonal functions. The planar functions are the result of a Lagrange interpolation for a number of design points.
5 Applied Mechanics and Materials Vol The sampling of the points and the building of the level curves were done by original computer programs written for AutoCAD in VB.NET. For the level curves, the program builds the Delaunay triangulation of the set of points, uses linear interpolation along the triangulation lines to get the points along the level curves and then connects those points to build the final level curves as AutoCAD spline entities. Although the level curves can be imported in any popular 3D modeling software to be used as primitives for the final model, we used AutoCAD modeling capabilities for this purpose. The end results were analyzed for precision with respect to each other and to the original model. References [1] F. Franceschini, D. Maisano, L. Mastrogiacomo, Mobile spatial coordinate measuring system (MScMS) and CMMs: a structured comparison, The International Journal of Advanced Manufacturing Technology, 2009, Volume 42, Issue 11-12, pp [2] G. Moroni, S. Petrò, Coordinate Measuring Machine Measurement Planning in Geometric Tolerances, Springer London, 2010, pp [3] S.W. Lin et al, Simulation of the errors transfer in an articulation-type coordinate measuring machine, The International Journal of Advanced Manufacturing Technology, 2006, Volume 30, Issue 9-10, pp [4] Z.C. Lin, Q.Y. Liu, Selection of coordinate measuring machines by the neural network method, 1997, Volume 13, Issue 1, pp [5] O. Hjelle, M. Daelen, Algorithms for Delaunay Triangulation in Triangulations and Applications, Springer Berlin Heidelberg, 2006, pp [6] M. de Berg, Computational Geometry: Algorithms and Applications, Springer, [7] R. H. J. Gmelig Meyling, Approximation by cubic C 1 -splines on arbitrary triangulations, Numerische Mathematik, 1987, Volume 51, Issue 1, pp [8] I. Mocian, R. Cazacu, New advances in application development for AutoCAD environment, Proceedings of the 9-th International Conference MTeM, 2009, pp [9] L.A. Zarrabeitia, V.H. Mederos, Multiresolution terrain modeling using level curve information, Journal of Computational and Applied Mathematics. 2013, Vol. 240, pp [10] D. Spinczyk, E. Pietka, Automatic Generation of 3D Lung Model, Computer Recognition Systems 2, Advances in Soft Computing, 2007, Volume 45, pp [11] VB.NET Documentation website (online), 2008, accessed 2012, available at: [12] O. Hjelle, M. Daelen, Programming Triangulations: The Triangulation Template Library (TTL) in Triangulations and Applications, Springer Berlin Heidelberg, 2006, pp [13] H. Edelsbrunner, N. R. Shah, Incremental topological flipping works for regular triangulations, Algorithmica, 1996, Volume 15, Issue 3, pp [14] R. Seidel, The upper bound theorem for polytopes: an easy proof of its asymptotic version, Computational Geometry, 1995, Volume 5, Issue 2, pp [15] P. Su, R.L.S. Drysdale, A Comparison of Sequential Delaunay Triangulation Algorithms, Proceedings of the eleventh annual symposium on Computational geometry, 1995, pp
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