Abdulwahed M. Abbas

Designing Catmull–Clark subdivision surfaces with curve interpolation constraints

Abstract Generating subdivision surfaces with curve interpolation constraints is needed in both computer graphics and geometric modeling applications. In the context of the Doo–Sabin subdivision scheme, this can be achieved through the use of polygonal complexes as suggested by Nasri (Presented at the Fifth Siam Conference on Geometric Design, Nashville, 1997; Comput. Aided Geom. Des. 17 (2000) 595). A polygonal complex is simply a polygonal mesh whose structure depends on the subdivision scheme used and whose limit of subdivision is a curve rather than a surface. The subdivision scheme applied to these complexes is basically the same applied to the mesh defining the surface but with possible modification of its subdivision rules. The advantage of that lies in the retention of the same subdivision coefficients, thus saving the need for any further analysis at the limit. In this paper, we propose a method for using polygonal complexes to generate Catmull–Clark subdivision surfaces with curve interpolation constraints. The polygonal complexes are embedded here in the given mesh, which can possibly interpolate intersecting curves.

Uniform B-Spline Curve Interpolation with Prescribed Tangent and Curvature Vectors

This paper presents a geometric algorithm for the generation of uniform cubic B-spline curves interpolating a sequence of data points under tangent and curvature vectors constraints. To satisfy these constraints, knot insertion is used to generate additional control points which are progressively repositioned using corresponding geometric rules. Compared to existing schemes, our approach is capable of handling plane as well as space curves, has local control, and avoids the solution of the typical linear system. The effectiveness of the proposed algorithm is illustrated through several comparative examples. Applications of the method in NC machining and shape design are also outlined.

Skinning Catmull-Clark subdivision surfaces with incompatible cross-sectional curves

One of the major problems in the generation of skinning B-spline surfaces is the incompatibility of the cross-sectional curves. This occurs when the cross sections are defined by control polygons having different number of control devices. Traditionally, this incompatibility is overcome by knot insertion that makes all control polygons have equal number of vertices. The drawback of this solution is that it can very quickly lead to an explosion in the number of vertices of the control mesh defining the skinning surface. In this paper, we show that this problem can be rectified through the use of subdivision surfaces. We describe an approach to generate a skinning Catmull-Clark subdivision surface through incompatible cross-sections of cubic B-spline curves. The resulting surface has all the properties of subdivision surfaces while requiring a smaller number of control points than those obtained through the more conventional techniques.

Generating B-spline curves with points, normals and curvature constraints: a constructive approach

This paper presents a constructive method for generating a uniform cubic B-spline curve interpolating a set of data points simultaneously controlled by normal and curvature constraints. By comparison, currently published methods have addressed one or two of those constraints (point, normal or cross-curvature interpolation), but not all three constraints simultaneously with C2 continuity. Combining these constraints provides better control of the generated curve in particular for feature curves on free-form surfaces. Our approach is local and provides exact interpolation of these constraints.

Lofted Catmull-Clark subdivision surfaces

One essential interpolation constraint on subdivision surfaces is curve interpolation. Subdivision surfaces through predefined meshes of curves can now be generated using either variations of existing subdivision schemes or (in our case) polygonal complexes. This paper goes one step further; given a sequence of cross sectional curves (c/sub i/), each defined by a uniform cubic B-spline control polygon (cp/sub i/), we present a technique for generating a lofted subdivision surface through these curves. The advantages of using polygonal complexes coupled with subdivision surfaces are that curves do not have to be compatible and that it is possible to locally control the cross curvature of a given cross section.

Constraint-based timetabling-a case study

This paper reports a case study in applying constraint satisfaction techniques to university and school timetabling. It involves the construction of a substantial, carefully specified, fully tested and fully operational system. The software engineering aspect of constraint satisfaction is emphasized. Constraint satisfaction problems are expressed in a language more familiar to the formal software engineering community. This brings constraint satisfaction one step closer to formal specification, program verification and transformation; issues extensively studied in software engineering. In problem formulation, explicit domain constraints and heuristic information are made explicit. Moreover, the users needs are considered more closely; for instance, when the program fails to find a solution, useful indications are produced to help in relaxation or reformulation of the problem.

Software engineering aspects of constraint-based timetabling—a case study

Abstract This paper details the stages of building a substantial, carefully specified, fully tested and fully operational university and school timetabling system. This is reported as a case study in applying Constraint Satisfaction techniques. The emphasis is on the software engineering aspects of the problem. That is, Constraint Satisfaction problems are expressed in a language more familiar to the formal software engineering community. Moreover, this language is used to formulate domain constraints and heuristic information. In addition to that, the users needs are looked at more closely. For instance, the system supplies indications useful for relaxing or reformulating the constraints of the problem when a solution satisfying these constraints is impossible to produce. This has a value in bringing Constraint Satisfaction one-step closer to formal specification, program verification and transformation.

Interpolating meshes of curves by Catmull-Clark subdivision surfaces with a shape parameter

This paper presents a solution to the problem of constructing Catmull-Clark subdivision surfaces interpolating given meshes of cubic B-spline curves. The solution is supplemented by an account of the continuity of the constructed surfaces at and around the extraordinary points of the corresponding meshes. This is conducted through an analysis of the corresponding subdivision matrix, which comes in parameterized form, thus adding a degree of freedom to the process. This degree of freedom gives room for manipulating the quality of the resulting surface without violating the initial interpolation constraints.

DEPICT: A High-Level Formal Language For Modeling Constraint Satisfaction Problems

The past decade witnessed rapid development of constraint satisfaction technologies, where algorithms are now able to cope with larger and harder problems. However, owing to the fact that constraints are inherently declarative, attention is quickly turning toward developing high-level programming languages within which such problems can be modeled and also solved. Along these lines, this paper presents DEPICT, the language. Its use is illustrated through modeling a number of benchmark examples. The paper continues with a description of a prototype system within which such models may be interpreted. The paper concludes with a description of a sample run of this interpreter showing how a problem modeled as such is typically solved.

T-Spline Polygonal Complexes

ABSTRACTThe notion of polygonal complexes was originally conceived as a means for exact interpolation of uniform B-spline curves by Doo-Sabin (and later on by Catmull-Clark) subdivision surfaces. Starting from the theoretical origin of these complexes, this paper provides a general formulation of this notion that covers all quad-based (uniform/non-uniform) B-spline as well as NURBS surfaces. This formulation is generalized even further to cope with the extra-requirements brought about in the context of T-spline surfaces while, at the same time, maintaining previous formulations as particular instances of that.