Hill, Steve and Roberts, Jonathan C.
Generating Surface Geometry in Higher Dimensions using Local Cell Tilers.
, Computing Laboratory, University of Kent, Canterbury, Kent CT2 7NF
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In two dimensions contour elements surround two dimensional objects, in three dimensions surfaces surround three dimensional objects and in four dimensions hypersurfaces surround hyperobjects. These surfaces can be represented by a collection of connected simplices, hence, continuous n dimensional surfaces can be represented by a lattice of connected n-1 dimensional simplices. The lattice of connected simplices can be calculated over a set of adjacent n-dimensional cubes, via for example the Marching Cubes Algorithm. These algorithms are often named local cell tilers. We propose that the local-cell tiling method can be usefully-applied to four dimensions and potentially to N-dimensions. We present an algorithm for the generation of major cases (cases that are topologically invariant under standard geometrical transformations) and introduce the notion of a sub-case which simplifies their representations. Each sub-case can be easily subdivided into simplices for rendering and we describe a backtracking tetrahedronization algorithm for the four dimensional case. An implementation for surfaces from the fourth dimension is presented and we describe and discuss ambiguities inherent within this and related algorithms.
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