Geometry for N-Dimensional Graphics

Textbook graphics treatments commonly use special notations for the geometry of 2 and 3 dimensions that are not obviously generalizable to higher dimensions. Here we collect a family of geometric formulas frequently used in graphics that are easily extendible to N dimensions as well as being helpful alternatives to standard 2D and 3D notations. What use are such formulas? In mathematical visualization, which commonly must deal with higher dimensions — 4 real dimensions, 2 complex dimensions, etc. — the utility is selfevident (see, e.g., (Banchoff 1990, Francis 1987, Hanson and Heng 1992b, Phillips et al. 1993)). The visualization of statistical data also frequently utilizes techniques of N -dimensional display (see, e.g., (Noll 1967, Feiner and Beshers 1990a, Feiner and Beshers 1990b, Brun et al. 1989, Hanson and Heng 1992a)). We hope that publicizing some of the basic techniques will encourage further exploitation of N -dimensional graphics in scientific visualization problems. We classify the formulas we present into the following categories: basic notation and the N -simplex; rotation formulas; imaging in N -dimensions;N -dimensional hyperplanes and volumes;N -dimensional cross-products and normals; clipping formulas; the point-hyperplane distance; barycentric coordinates and parametric hyperplanes; N -dimensional ray-tracing methods. An appendix collects a set of obscure Levi-Civita symbol techniques for computing with determinants. For additional details and insights, we refer the reader to classic sources such as (Sommerville 1958, Coxeter 1991, Hocking and Young 1961) and (Banchoff and Werner 1983, Efimov and Rozendorn 1975).