Marching cubes: A high resolution 3D surface construction algorithm
SIGGRAPH '87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques
Interval analysis for computer graphics
SIGGRAPH '92 Proceedings of the 19th annual conference on Computer graphics and interactive techniques
Distance approximations for rasterizing implicit curves
ACM Transactions on Graphics (TOG)
An efficient method for analyzing the topology of plane real algebraic curves
Selected papers presented at the international IMACS symposium on Symbolic computation, new trends and developments
Guaranteeing the topology of an implicit surface polygonization for interactive modeling
Proceedings of the 24th annual conference on Computer graphics and interactive techniques
Dual contouring of hermite data
Proceedings of the 29th annual conference on Computer graphics and interactive techniques
Rasterizing Algebraic Curves and Surfaces
IEEE Computer Graphics and Applications
Comparison of interval methods for plotting algebraic curves
Computer Aided Geometric Design
Sampling and meshing a surface with guaranteed topology and geometry
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Complete, exact, and efficient computations with cubic curves
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Isotopic implicit surface meshing
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Isotopic approximation of implicit curves and surfaces
Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing
On the exact computation of the topology of real algebraic curves
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Provably good sampling and meshing of surfaces
Graphical Models - Solid modeling theory and applications
Effective Computational Geometry for Curves and Surfaces (Mathematics and Visualization)
Effective Computational Geometry for Curves and Surfaces (Mathematics and Visualization)
Complete subdivision algorithms, II: isotopic meshing of singular algebraic curves
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
An exact and efficient approach for computing a cell in an arrangement of quadrics
Computational Geometry: Theory and Applications - Special issue on robust geometric algorithms and their implementations
Topologically accurate meshing using domain subdivision techniques
Topologically accurate meshing using domain subdivision techniques
Discrete & Computational Geometry - Special Issue: 25th Annual Symposium on Computational Geometry; Guest Editor: John Hershberger
A simple but exact and efficient algorithm for complex root isolation
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Empirical study of an evaluation-based subdivision algorithm for complex root isolation
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Adaptive isotopic approximation of nonsingular curves and surfaces
Adaptive isotopic approximation of nonsingular curves and surfaces
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We consider the problem of approximating nonsingular surfaces which are implicitly represented by equations of the form f(x,y,z)=0. Our correctness criterion is an isotopy of the approximate surface to the exact surface. We focus on methods based on domain subdivision using numerical primitives. Such methods are practical and have adaptive and local complexity. Previously, Snyder (1992) [3] and Plantinga-Vegter (2004) [4] have introduced techniques based on parameterizability and non-local isotopy, respectively. In our previous work (SoCG 2009), we synthesized these two techniques into an efficient and practical algorithm for curves. This paper extends our approach to surfaces. The extension is by no means routine: the correctness argument is much more intricate. Unlike the 2-D case, a new phenomenon arises in which local rules for constructing surfaces are no longer sufficient. We treat an important extension to exploit anisotropic subdivision. Anisotropy means that we allow boxes to be split into 2, 4 or 8 subboxes with arbitrary but bounded aspect ratio. This could greatly improve the adaptivity of the algorithm. Our algorithms are relatively easy to implement, as the underlying primitives are based on interval arithmetic and exact BigFloat numbers. We report on encouraging preliminary experimental results.