Transformation and Normal Vector Calculation of Parametrically Defined Surfaces Based on Dual Vectors and Screw Theory: Application to Phongʹs Shading Model

This paper presents a new approach for the transformation and normal vector calculation algorithms of parametrically defined surfaces via dual vectors and line transformations. The surface is defined via dual points, the transformation is performed by rotations and translations based on screw theory while normal vector calculation is utilized for shading based on Phong’s illumination model. The main benefit of this approach lies into the compactness of the surface’s representation since geometrical characteristics, such as tangent vectors, that are necessary for shading algorithms, are included within its definition. An extensive comparison is performed between the proposed approach and the traditional homogeneous model, presenting the merits of our approach. Analytical and experimental determination of the computational cost via computer implementation of 3D surface transformation and shading is presented. Point-based methods for the representation, transformation and shading of parametrically defined surfaces are compared to the introduced line-based methods (dual quaternions and dual orthogonal matrices). It is shown that the simplified rendering procedure of 3D objects, is considerably faster using screw theory over the traditional point-based structures