Coordinate relationships between vertices of linear octree nodes and corners of the universe

Abstract The display and manipulation of octree-encoded objects require combined translation-rotation operations in order to transform the object representation from the world coordinate system (WCS) to the viewing coordinate system (VCS). These types of operations are often very expensive in terms of execution time when they are performed on every octree node in the raster. This article provides an explicit formula which computes the coordinates of any vertex of any linear octree node in terms of coordinates of the corners of the universe in which it is enclosed. This formula is invariant under all affine transformation operations that operate on the entire object space. Furthermore, it only requires additions and multiplications by powers of 2. When performing object manipulations or display operations, the standard matrix transformations (translations, rotations, and scaling) need only be applied once to the main vertices of the universe cube. The coordinates of the vertices of individual object elements are determined by using the formula introduced in this article.