====== Isosurface Polygonization using Hyperbolic Arcs to Resolve Topological Ambiguity Problems ======
Pasko A. A., Pilyugin V. V., Pokrovskiy V. N.,\\
"Geometric modeling in the analysis of trivariate functions",\\
Computers and Graphics, vol.12, Nos.3/4, 1988, pp.457-465.\\
Electronic version: [[http://hyperfun.org/C&G88.pdf|scanned, PDF (3.6Mb)]]

It is an English version of\\
Pasko A. A., Pilyugin V. V., Pokrovskiy V. N., Using computer geometry for analysis of functions of three variables, Communications of Joint Institute of Nuclear Research, JINR P10-86-310, Dubna, Russia, 1986, 10 p. (in Russian).\\
Electronic version: [[http://hyperfun.org/P10-86-310.pdf|PDF (298K)]] scanned by the [[http://www.slac.stanford.edu/spires/find/hep/www?indexer=1&rawcmd=find+r+JINR-P10-86-310|SPIRES HEP]] project (U. Stanford).




===== Abstract =====
An algorithm for polygonization of an isosurface of a function of three variables is presented. The trilinear interpolation inside a cubic cell and the bilinear interpolation on a cell face are used for the hyperbolic arcs detection at the faces of a cell and for the construction of the edges connectivity graph to resolve possible topological ambiguities.

{{frep:poly_box.jpg?400nolink}}**a** {{frep:poly_patch.jpg?400nolink}}**b** \\
{{frep:poly_graph.jpg?400nolink}}**c** {{frep:poly_arc.jpg?400nolink}}**d**

Algorithm for isosurface polygonization using hyperbolic arcs:\\
a) subdivision of a bounding box into rectangular cells;\\
b) example of surface patches inside a cell with hyperbolic arcs on the cell faces;\\
c) cycles in the connectivity graph corresponding to surface patches in the cell;\\
d) the topological ambiguity case of four intersection points of cell face edges with the isosurface is resolved using hyperbolic arcs (resulting from the trilinear interpolation inside the cell and the bilinear interpolation on the cell face).

This algorithm is implemented in the [[hyperfun:tools|HyperFun Polygonizer]].\\
See examples of polygonized isosurfaces at the [[gallery:main|HyperFun Gallery]].

See the related topics:\\
[[sharp|Sharp Features of Polygonized Implicit Surfaces]]\\
[[sharp_em|Polygonization with Embedded Sharp Features Extraction]]\\
[[discret|Surface and Volume Discretization of Functionally Based Heterogeneous Objects]]\\
[[trimming|Trimmimg Implicit Surfaces]]\\