A Hidden-Line Algorithm for Hyperspace

An object-space hidden-line algorithm for higher-dimensional scenes has been designed and implemented. Scenes consist of convex hulls of any dimension, each of which is compared against the edges of all convex hulls not eliminated by a hyperdimensional clipper, a depth test after sorting and a minimax text. Hidden and visible elements are determined in accordance with the dimensionality of the selected viewing hyperspace. When shape alone is the attribute of interest, hidden-line elimination need be performed only in that hyperspace. The algorithm is of value in the production of shadows of hyperdimensional models, including but not limited to four-dimensional space-time models, the hyperdimensional elementary catastrophe models and multivariate statistical models. Key words, hyperspace, hidden-line elimination Introduction. This paper describes an algorithm for solving the hidden-line problem in hyperspace. The lines are edges of convex hulls approximating the surfaces of hyperdimensional objects. The algorithm removes edges which would be invisible in a hyperdimensional scene. The scene may then be projected to lower dimensions. The development of a hidden-line eliminator for hyperspace is part of an ongoing effort to display and gain insight into the structures of higher-dimensional space. Of particular interest are four-dimensional space-time models, the seven elementary catastrophe models, of which five are hyperdimensional [1] and multivariate statistical models. While these models are numerically useful to some extent, they are of limited general utility in the absence of adequate tiyperdimensional presentation techniques. Without an ability to present visible lines only, the four(and higher-) dimensional analogues of front, rear and depth become hopelessly garbled in the generalized view. Previous efforts to display hyperobjects [2], 3] have employed techniques which either discard one or more variables or hold them constant so as to restrict structures to three dimensions. Such techniques impose unacceptable constraints. To illustrate by analogy, consider a cube aligned with x-, yand z-axes. The cube can be restricted to two dimensions by either eliminating or holding constant one of the coordinates. The cube then appears to be nothing more than a square (see Fig. l a). A generalized technique, which permits the cube to be viewed from any position, with any orientation and in stereo, provides substantially more information, especially when hidden elements of the cube are removed (see Fig. lb). Similarly, views of hyperobjects are significantly enriched when a generalized viewing capability is combined with a hyperdimensional hidden-line eliminator such as the one described in this paper.