Fast reliable interrogation of procedurally defined implicit surfaces using extended revised affine arithmetic.

Techniques based on interval and previous termaffine arithmetic next term and their modifications are shown to provide previous term reliable next term function range evaluation for the purposes of previous termsurface interrogation.next term In this paper we present a technique for the previous termreliable interrogation of implicit surfacesnext term using a modification of previous termaffine arithmeticnext term called previous term revised affine arithmetic.next term We extend the range of functions presented in previous termrevised affine arithmeticnext term by introducing previous termaffinenext term operations for arbitrary functions such as set-theoretic operations with R-functions, blending and conditional operators. The obtained previous termaffinenext term forms of arbitrary functions provide previous termfasternext term and tighter function range evaluation. Several case studies for operations using previous termaffinenext term forms are presented. The proposed techniques for previous termsurface interrogationnext term are tested using ray-previous termsurfacenext term intersection for ray-tracing and spatial cell enumeration for polygonisation. These applications with our extensions provide previous termfast and reliablenext term rendering of a wide range of arbitrary previous termprocedurally defined implicit surfacesnext term (including polynomial previous termsurfaces,next term constructive solids, pseudo-random objects, previous termprocedurally definednext term microstructures, and others). We compare the function range evaluation technique based on previous termextended revised affine arithmeticnext term with other previous termreliablenext term techniques based on interval and previous termaffine arithmeticnext term to show that our technique provides the previous termfastestnext term and tightest function range evaluation for previous termfast and reliable interrogation of procedurally defined implicit surfaces.next term Research Highlights The main contributions of this paper are as follows. ► The widening of the scope of previous termreliablenext term ray-tracing and spatial enumeration algorithms for previous termsurfacesnext term ranging from algebraic previous termsurfaces (definednext term by polynomials) to general previous termimplicit surfaces (definednext term by function evaluation procedures involving both previous termaffinenext term and non-previous termaffinenext term operations based on previous termrevised affine arithmetic)next term. ► The introduction of a technique for representing procedural models using special previous termaffinenext term forms (illustrated by case studies of previous termaffinenext term forms for set-theoretic operations in the form of R-functions, blending operations and conditional operations). ► The detailed derivation of special previous termaffinenext term forms for arbitrary operators

Efficient contouring of functionally represented objects for additive manufacturing

Functionally (implicitly) defined 3D objects allow us to quite easily model parts with complex topology such as lattices and organic-like structures with a high level of flexibility. Previous works in this area are based on the direct generation of CNC programs for the 3D printing of these objects and are backed by the growing support for this input format from hardware manufacturers. Efficient contouring of functionally defined models, however, is not an easy task. In this paper, we develop an algorithm for contour extraction of implicitly defined objects for direct additive manufacturing (AM). By comparing various adaptive and exhaustive (non-adaptive) methods of the function representation contouring for AM (FRepCAM), we make a set of recommendations for its usage depending on the specific resolution of the printer. In particular, we use a novel criterion based on affine arithmetic to maintain efficiency while preserving the robustness of the contouring process. The techniques mentioned were evaluated for algebraic and non-algebraic solids and heterogeneous models under a resolution that is comparable with that of current AM technology. The results show that the chosen adaptation criteria allow us to efficiently obtain a contour for complex models and generally outperform those of traditional algorithms based on exhaustive enumeration, especially for high-resolution contouring. In addition, the results present proof of the printability of implicitly defined objects with different 3D printing techniques

Implicit Linear Interval Estimations

Visualization and collision detection are two of the most important problems connected with implicit objects. Enumeration algorithms can be used either directly or as preprocessing step for many algorithms solving these problems. In general, enumeration algorithms based on recursive space subdivision are reliable tools to encounter those parts in space, where the object might be located. But the bad performance and the huge number of computed enclosing cells, if high precision is required, are grave drawbacks. Implicit Linear Interval Estimations (ILIEs) introduced in this paper are implicit interval (hyper-)planes providing oriented tight bounds of the object within given cells. It turns out that the use of ILIEs highly improves the performance of the classical enumeration algorithm and the quality of the results. The theoretical background as well as a fast and simple technique to compute ILIEs are presented. The applicability of ILIEs is demonstrated by means of a modified enumeration algorithm that has been implemented and tested for implicit surfaces

Approximating Implicit Curves On Triangulations With Affine Arithmetic

We present an adaptive method for computing a robust polygonal approximation of an implicit curve in the plane that uses affine arithmetic to identify regions where the curve lies inside a thin strip. Unlike other interval methods, even those based on affine arithmetic, our method works on triangulations, not only on rectangular quad trees. © 2012 IEEE.94101De Figueiredo, L.H., Stolfi, J., Affine arithmetic: Concepts and applications (2004) Numerical Algorithms, 37 (1-4 SPEC. ISS.), pp. 147-158. , DOI 10.1023/B:NUMA.0000049462.70970.b6Dobkin, D.P., Levy, S.V.F., Thurston, W.P., Wilks, A.R., Contour tracing by piecewise linear approximations (1990) ACM Transactions on Graphics, 9 (4), pp. 389-423Persiano, R.C.M., Ao, J., Comba, L.D., Barbalho, V., An adaptive triangulation refinement scheme and construction (1993) Proceedings of SIBGRAPI'93, pp. 259-266Suffern, K.G., Fackerell, E.D., Interval methods in computer graphics (1991) Computers & Graphics, 15 (3), pp. 331-340Mitchell, D.P., Three applications of interval analysis in computer graphics (1991) Frontiers in Rendering Course Notes. SIGGRAPH'91, pp. 1401-1413Lopes, H., Oliveira, J.B., De Figueiredo, L.H., Robust adaptive polygonal approximation of implicit curves (2002) Computers & Graphics, 26 (6), pp. 841-852Comba, J.L.D., Stolfi, J., Affine arithmetic and its applications to computer graphics (1993) Proceedings of SIBGRAPI'93, pp. 9-18Martin, R., Shou, H., Voiculescu, I., Bowyer, A., Wang, G., Comparison of interval methods for plotting algebraic curves (2002) Computer Aided Geometric Design, 19 (7), pp. 553-587. , DOI 10.1016/S0167-8396(02)00146-2, PII S0167839602001462De Cusatis Jr., A., De Figueiredo, L.H., Gattass, M., Interval methods for ray casting implicit surfaces with affine arithmetic (1999) Proceedings of SIBGRAPI'99, pp. 65-71. , IEEE PressDe Figueiredo, L.H., Stolfi, J., Velho, L., Approximating parametric curves with strip trees using affine arithmetic (2003) Computer Graphics Forum, 22 (2), pp. 171-179Bühler, K., Fast and reliable plotting of implicit curves (2002) Uncertainty Geometric Computations, pp. 15-28. , Kluwer AcademicBühler, K., Implicit linear interval estimations (2002) Proceedings of SCCG '02, pp. 123-132. , ACMMoore, R.E., (1966) Interval Analysis, , Prentice-HallStolfi, J., De Figueiredo, L.H., (1997) Self-Validated Numerical Methods and Applications, , 21st Brazilian Mathematics Colloquium, IMPATaubin, G., Rasterizing algebraic curves and surfaces (1994) IEEE Computer Graphics and Applications, 14, pp. 14-23Velho, L., Zorin, D., 4-8 Subdivision (2001) Computer Aided Geometric Design, 18 (5), pp. 397-427. , DOI 10.1016/S0167-8396(01)00039-5, PII S0167839601000395Kobbelt, L., Radic 3-subdivision (2000) Proceedings of SIGGRAPH '00, pp. 103-112. , AC