Piecewise linear hypersurfaces using the Marching Cubes Algorithm

Surface visualization is very important within scientific visualization. The surfaces depict a value of equal density (an isosurface) or display the surrounds of specified objects within the data. Likewise, in two dimensions contour plots may be used to display the information. Thus similarly, in four dimensions hypersurfaces may be formed around hyperobjects. These surfaces (or contours) are often formed from a set of connected triangles (or lines). These piecewise segments represent the simplest non-degenerate object of that dimension and are named simplices. In four dimensions a simplex is represented by a tetrahedron, which is also known as a 3-simplex. Thus, a continuous n dimensional surface may be represented by a lattice of connected n-1 dimensional simplices. This lattice of connected simplices may be calculated over a set of adjacent n dimensional cubes, via for example the Marching Cubes Algorithm. We propose that the methods of this local-cell tiling method may be usefully-applied to four dimensions and potentially to N-dimensions. Thus, we organise the large number of traversal cases and major cases;: introduce the notion of a sub-case (that enables the large:number of cases to be further reduced); and describe three methods for implementing the Marching Cubes lookup table in four-dimensions.