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Part II: Discrete ModellingMinimally Thin Discrete TriangulationValentin E. Brimkov, Reneta P. Barneva and Philippe NehligIn this chapter we study the possibility of obtaining a "minimally thin" 6tunnelfree voxelisation of a mesh of triangles. We propose an efficient algorithm for discretisation of a triangle and triangular meshes. The triangle sides and its interior are approximated by a special kind of discrete lines and planes that can be considered as maximally thin, and thus the obtained mesh voxelisation appears to be optimally thin as well. The algorithm uses integer arithmetic and its time complexity is linear in the number of the generated voxels. The discrete triangles admit a simple analytical description, which may provide certain advantages. Some initial experiments have shown that the quality of the obtained voxelisation is quite satisfactory. For relevant information, please visitSmooth Boundary Surfaces from Binary 3D DatasetsDaniel CohenOr, Arie Kadosh, David Levin and Roni YagelThe finite resolution of discrete surfaces causes various artifacts when one tries to render the surface from close or to zoomin. In this chapter we show that the use of a high order interpolators greatly enhances ones ability to display discrete surfaces from up close. We show that trilinear interpolation is not smooth enough and introduce a tricubic reconstruction based on hermit interpolation. Manufacturing IsovolumesMichael BaileyIsosurfaces are an effective way to visualise the distribution of scalar data values in a volume. Attaching the isovalue to an interactive input, such as a slider bar or a dial, provides further insight as one watches the isosurfaces change shape and meld into each other. Unfortunately, such interaction is fleeting. Once one has passed on to a new isovalue transition, it is difficult to remember what a previous one looked like. There is a significant amount of insight to be gained by examining the relationship between different isosurface shapes, but this is hard to do on a graphics screen. This chapter presents an approach to solving this problem by manufacturing interlocking isovolumes. This allows a series of isosurfaces to be seen simultaneously and their shapes compared using visual and touch feedback. This chapter shows how the standard Marching Cubes algorithm was extended to accomplish this and shows an example. 
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