Drawing ruled surfaces using the Dual de Boor algorithm

This paper outlines emerging methods, presently under investigation, for constructing ruled surfaces and solid-body trajectories in a Computer Aided Geometric Design (CAGD) context. The underlying principle is that of the standard De Boor Algorithm [4,2,15] for constructing B-spline curves on the plane, or in 3-dimensional space. But the coordinates of all 3-vector points including the control points to be interpolated are now taken to be dual, not real, numbers. When the six resulting number components are interpreted as Pl ű cker coordinates [12], each dual point finds interpretation as a line, and each curve as a ruled surface, in 3-dimensional space. The construction of a duplicated surface is discussed along with some new methods based on a rigid body character. The problem of drawing infinitely long lines, on a screen, as represented by Pl űcker coordinates, is touched upon.Rena Din

Dual space drawing methods of cylinders

Ruled surface design and its applications are widely used in CAGD (Computer Aid Geometric Design) and industrial areas. The cylinder is a particular ruled surface. Its drawing methods lay the foundation for drawing other ruled surfaces, such as cones, with dual space drawing methods. In this paper, new methods are introduced for drawing cylinders. The underlying principle of these methods is the application and extension (variation) of the dual De Casteljau algorithm [11]. Based on this, a comprehensive study is carried out for the creation of cylinders. The dual De Casteljau algorithm is adjusted by adding boundary conditions to draw cylinders with smooth closed trajectories. The dual space drawing methods of the circular cylinder with boundary conditions and segment construction are also discussed.Rena Din

The extension of the dual De Casteljau algorithm

© Copyright 2003 IEEERuled surface design and its applications are widely used in CAGD (computer aided geometric design) and industrial areas. The dual De Casteljau algorithm [K. Sprott et. al., (1997)] is for drawing ruled surfaces. However, it cannot draw any type of ruled surfaces. Here, the algorithm is extended to draw the hyperboloid which could be indefinite if it is drawn by the initial algorithm. The dual characteristics of ruled surfaces created by using the dual method are investigated. The special displacement method of the rulings with two parallel control screws is also discussed. The extension of the dual De Casteljau algorithm has shown the wider usage of the dual approach for drawing ruled surfaces

Model Updating CTL Systems

The original publication can be found at www.springerlink.comMinimal change is a fundamental principle for modelling system dynamics. In this paper, we study the issue of minimal change for Computational Tree Logic (CTL) model update. We first consider five primitive updates which capture the basic update operations in the CTL model. Based on these primitive updates, we then define the minimal change criteria for CTL model update and develop formal algorithms that embed the underlying minimal change principle. We also present the well known microwave oven scenario to demonstrate our update algorithms. Our work presented in this paper can be viewed as the first formalization towards an integration of model checking and model updating for system modification.Yulin Ding and Yan Zhan

System modification case studies

© Copyright 2007 IEE

Algorithms for CTL System Modification

The original publication can be found at www.springerlink.comModel updating, as a new concept to be employed as a standard and universal method for system modification, has been started in [4] and further developed in this paper. This paper introduces algorithms for Computation Tree Logic (CTL) model update. The algorithms correct errors in a CTL Kripke model to make this model satisfy its various required properties. These update algorithms are designed through pseudo-code, which details the logic of the algorithms. The microwave oven example in [3] has been correctly treated by the pseudo-code, which demonstrates the feasibility of these algorithms. The algorithms and their pseudo-code are the foundations for later implementation of a CTL model updater and integration of a CTL model checker and this updater.Yulin Ding and Yan Zhan

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.Rena Ding, Yan Zhan

CTL model update: semantics, computations and implementation

http://trove.nla.gov.au/work/2580336

Dual space drawing methods for cones

Ruled surface design and its applications are widely used in CAGD (Computer Aid Geometric Design) and industrial areas. In this paper, the cone is a particular ruled surface, which is drawn by using the dual methods based on the cylindrical shape [Ding 2003] with its same base curve. The underlying principle is the conversion from cylinders to cones by application of the dual displacement methods [Sprott and Ravani 1997]. Based on the dual methods, a comprehensive study is carried out for the creation of cones. The translation, which satisfies the condition of dual methods, from cylinders which have been created by using the dual De Casteljau algorithm [Ding 2003] to cones is investigated. During the process, a new theorem, the tangential interpolation line theorem is unveiled. This discovery lays the foundation for the conversion from cylinders to cones. The new methods are put forward to draw different cones, which have been successfully implemented by computer simulation.Rena Ding and Yan Zhan

The dual drawing method of the hyperbolic paraboloid and the screen representation of the ruling

Ruled surface design and its applications are widely used in CAGD(Computer Aid Geometric Design) and industrial areas. In this paper, the hyperbolic paraboloid is a particular ruled surface, which is drawn by using the dual De Casteljau algorithm [15]. The characteristics of the hyperbolic paraboloid are analysed in order to match the requirement of the dual De Casteljau algorithm. A new method for the screen representation of the ruling is introduced. The new method presents the ruling as a clipped line segment on the screen in order to show the proportion of the ruled surface on the screen. The two new methods represent the universal approach to draw ruled surfaces by a combination of the theoretical and practical aspects of dual methods.Rena Ding and Yan Zhan