Computer graphics representation and transformation of geometric entities using dual unit vectors and line transformations

Abstract

In this paper, a representational model is proposed for the description and transformation of three-dimensional geometric entities in computer graphics. The structure of the proposed representation is based on dual unit vectors, while the corresponding transformations are carried out through dual unit quaternions or dual orthogonal matrices. The main advantage of this representation is its compactness since the additional useful geometric characteristics of a represented curve or surface such as a tangent or normal vector are incorporated within the actual representational structure itself. Rotations, translations and view transformations are naturally expressed using the concept of screw displacement, while scaling is accomplished utilizing the moment vector of each dual line. Furthermore, an analysis of the transform operator based on dual unit quaternions is presented in order to ascertain an efficient formula to be used in the implementation of a computational algorithm for computer animation. Finally, an analytical comparison between the proposed representational model and the usual homogeneous model in computer animation is presented showing the merits of our method.