Dual Space Drawing Methods for Ruled Surfaces with Particular Shapes

Ruled surface design and its applications are widely used in CAGD (Computer Aided Geometric Design) and industrial areas. The hyperbolic paraboloid, the cylinder, the cone, and the hyperboloid are ruled surfaces with particular shapes. In this paper, dual space drawing methods for these particular ruled surfaces are proposed. The underlying principles of these methods are the applications, extensions and variations of the dual (space) de Casteljau algorithm [15].The hyperbolic paraboloid is created by using the initial dual de Casteljau algorithm directly. The cylinder, the cone, and the hyperboloid, on the other hand, are created by using the extensions or variations of the dual de Casteljau algorithm. During the drawing processes, the screen representation of the rulings is presented. This treatment presents the rulings as clipped line segments on the computer screen in order to show the proportion of the ruled surface on the computer screen. In addition, the dual characteristics of Bezier surfaces are also discussed. The special construction method of the displacement matrix with two parallel control screws is demonstrated. The normal screw is normalized into the unit screw as well. These new methods show the universal possibility of the dual space approach for drawing ruled surfaces.