Abstract
We describe the algebraic-geometric modeling platform Axl, which provides tools for the manipulation, computation and visualisation of semi-algebraic models. This includes meshes, basic geometric objects such as spheres, cylinders, cones, ellipsoids, torus, piecewise polynomial parameterisations of curves, surfaces or volumes such as b-spline parameterisations, as well as algebraic curves and surfaces defined by polynomial equations. Moreover, Axl provides algorithms for processing these geometric representations, such as computing intersection loci (points, curves) of parametric models, singularities of algebraic curves or surfaces, certified topology of curves and surfaces, etc.
We present its main features and describe its generic extension mechanism, which allows one to define new data types and new processes on the data, which benefit from automatic visualisation and interaction facilities. The application capacities of the software are illustrated by short descriptions of plugins on algebraic curves and surfaces and on splines for Isogeometric Analysis.
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References
Alberti, L., Mourrain, B.: Regularity criteria for the topology of algebraic curves and surfaces. In: Martin, R., Sabin, M., Winkler, J. (eds.) Mathematics of Surfaces XII. LNCS, vol. 4647, pp. 1–28. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-73843-5_1
Alberti, L., Mourrain, B.: Visualisation of implicit algebraic curves. In: Pacific Conference on Computer Graphics and Applications, Lahaina, Maui, Hawaii, United States, pp. 303–312. IEEE Computer Society, October 2007
Alberti, L., Mourrain, B., Técourt, J.P.: Isotopic triangulation of a real algebraic surface. J. Symb. Comput. 44(9), 1291–1310 (2009)
Alberti, L., Mourrain, B., Wintz, J.: Topology and arrangement computation of semi-algebraic planar curves. Comput. Aided Geom. Des. 25(8), 631–651 (2008)
Emiris, I., Mantzaflaris, A., Mourrain, B.: Voronoi diagrams of algebraic distance fields. Comput. Aided Des. 45(2), 511–516 (2013)
Giannelli, C., Juettler, B., Kleiss, S.K., Mantzaflaris, A., Simeon, B., Speh, J.: THB-splines: an effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis. Comput. Methods Appl. Mech. Eng. 299, 337–365 (2016)
Juettler, B., Langer, U., Mantzaflaris, A., Moore, S., Zulehner, W.: Geometry + simulation modules: implementing isogeometric analysis. Proc. Appl. Math. Mech. 14(1), 961–962 (2014)
Langer, U., Mantzaflaris, A., Moore, S.E., Toulopoulos, I.: Multipatch discontinuous galerkin isogeometric analysis. In: Jüttler, B., Simeon, B. (eds.) Isogeometric Analysis and Applications 2014. LNCSE, vol. 107, pp. 1–32. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-23315-4_1
Liang, C., Mourrain, B., Pavone, J.P.: Subdivision methods for the topology of 2D and 3D implicit curves. In: Juetller, B., Piene, R. (eds.) Geometric Modeling and Algebraic Geometry, pp. 199–214. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-72185-7_11
Mantzaflaris, A., Mourrain, B.: A subdivision approach to planar semi-algebraic sets. In: Mourrain, B., Schaefer, S., Xu, G. (eds.) GMP 2010. LNCS, vol. 6130, pp. 104–123. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13411-1_8
Mourrain, B., Pavone, J.P.: Subdivision methods for solving polynomial equations. J. Symb. Comput. 44(3), 292–306 (2009)
Schroeder, W., Martin, K., Lorensen, B.: The Visualization Toolkit, 4th edn. Kitware, Clifton Park (2006)
Wintz, J., Kloczko, T., Niclausse, N., Rey, D.: dtk - a metaplatform for scientific software development. ERCIM News 2012(88) (2012). http://ercim-news.ercim.eu/en88/ri/dtk-a-metaplatform-for-scientific-software-development
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Christoforou, E., Mantzaflaris, A., Mourrain, B., Wintz, J. (2018). Axl, a Geometric Modeler for Semi-algebraic Shapes. In: Davenport, J., Kauers, M., Labahn, G., Urban, J. (eds) Mathematical Software – ICMS 2018. ICMS 2018. Lecture Notes in Computer Science(), vol 10931. Springer, Cham. https://doi.org/10.1007/978-3-319-96418-8_16
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DOI: https://doi.org/10.1007/978-3-319-96418-8_16
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