CONSTRAINT MODELING FOR CURVES AND SURFACES IN CAGD: A SURVEY
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paper_title: From on-line sketching to 2D and 3D geometry: a system based on fuzzy knowledge

paper_content:

The paper describes the development of a fuzzy knowledge-based prototype system for conceptual design. This real time system is designed to infer user's sketching intentions, to segment sketched input and generate corresponding geometric primitives: straight lines, circles; arcs, ellipses, elliptical arcs, and B-spline curves. Topology information (connectivity, unitary constraints and pairwise constraints) is received dynamically from 2D sketched input and primitives. From the 2D topology information, a more accurate 2D geometry can be built up by applying a 2D geometric constraint solver. Subsequently, 3D geometry can be received feature by feature incrementally. Each feature can be recognised by inference knowledge in terms of matching its 2D primitive configurations and connection relationships. The system accepts not only sketched input, working as an automatic design tool, but also accepts user interactive input of both 2D primitives and special positional 3D primitives. This makes it easy and friendly to use. The system has been tested with a number of sketched inputs of 2D and 3D geometry.
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paper_title: Integrating constraints and direct manipulation

paper_content:

In this paper, we present techniques for integrating constraint and direct manipulation approaches to geometric modeling. Direct manipulation positioning techniques are augmented to provide the option of making the relationships they establish persistent. Differential constraint techniques are used to maintain these relationships during subsequent editing. Issues in displaying and editing constraints are also addressed. By integrating constraints with direct manipulation, it is possible to build systems that provide the power of explicit representation of geometric relationships and the properties which make direct manipulation so attractive.
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paper_title: Constraint-based curve manipulation

paper_content:

Alternative techniques for control-vertex manipulation of parametric curves are described. Their generality and flexibility make possible a wide range of curve-editing interfaces based on direct manipulation. A method for applying systems of constraints at one or more arbitrary points on a single or composite curve and an efficient technique for the interactive application of these systems are outlined. Analytic solutions and examples for useful systems of constraints are presented. >
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paper_title: A Free Form Feature Taxonomy

paper_content:

In this paper the notion of free form feature for aesthetic design is presented. The design of industrial products constituted by free form surfaces is done by using CAD systems representing curves and surfaces by means of NURBS functions, which are usually defined by low level entities that are not intuitive and require some knowledge of the mathematical language. Similarly to the feature-based approach adopted by CAD systems for classical mechanical design, a set of high level modelling entities which provides commonly performed shape modifications has been identified. Particularly, the paper suggests a classification of the so-called detail features for an aesthetic and/or functional characterization of predefined free form surfaces. Feature types are formally described by means of an analytical definition of the surface modification through deformation and elimination laws. A topological classification is then given according to the application domain of such laws. A further sub-classification of morphological types is then suggested according to geometric properties of weak convexity and concavity for the resulting modified shape, leading to a taxonomy of simple free form features meaningful for aesthetic design.
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paper_title: Specification of freeform features

paper_content:

Freeform feature modeling is an extension to feature modeling in which, in addition to regular-shaped features, also freeform features are possible. Due to the large variety of freeform features, a generic approach to specify such features is required. This paper describes such an approach. A freeform feature class is specified by using a prototype and constraints. The latter are used to define intuitive parameters and validity conditions for the class. A new, prototype-driven constraint solving method is used to unambiguously determine a freeform feature during the specification both of a class and of an instance to be added to a model. The specification of freeform features and the prototype-driven constraint solving method are discussed. Several examples are given.
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paper_title: Hierarchical B-spline refinement

paper_content:

Refinement is usually advocated as a means of gaining finer control over a spline curve or surface during editing. For curves, refinement is a local process. It permits the change of control vertices and subsequent editing of the detail in one region of the curve while leaving control vertices in other regions unaffected. For tensor-product surfaces, however, refinement is not local in the sense that it causes control vertices far from a region of interest to change as well as changing the control vertices that influence the region. However, with some care and understanding it is possible to restrict the influence of refinement to the locality at which an editing effect is desired. We present a method of localizing the effect of refinement through the use of overlays, which are hierarchically controlled subdivisions. We also introduce two editing techniques that are effective when using overlays: one is direct surface manipulation through the use of edit points and the other is offset referencing of control vertices.
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paper_title: Simple constrained deformations for geometric modeling and interactive design

paper_content:

Deformations are a powerful tool for shape modeling and design. We present a new model for producing controlled spatial deformations, which we term Simple Constrained Deformations (Scodef) . The user defines a set of constraint points, giving a desired displacement and radius of influence for each. Each constraint point determines a local B-spline basis function centered at the constraint point, falling to zero for points beyond the radius. The deformed image of any point in space is a blend of these basis functions, using a projection matrix computed to satisfy the constraints. The deformation operates on the whole space regardless of the representation of the objects embedded inside the space. The constraints directly influence the final shape of the deformed objects, and this shape can be fine-tuned by adjusting the radius of influence of each constraint point. The computations required by the technique can be done very efficiently, and real-time interactive deformation editing on current workstations is possible.
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paper_title: Better Pasting Via Quasi-Interpolation

paper_content:

Abstract : Surface pasting is a hierarchical modeling technique that places feature surface atop base surface with approximate continuity. In this paper, we describe the use of quasi-interpolation to set the boundary control points of the pasted feature. To achieve interactive performance for surface pasting, we modify the coefficients and weights of quasi-interpolation and devise an efficient sampling scheme. The new surface pasting technique is actually faster than the original surface pasting method when using the same number of feature control points, with a much lower discontinuity between the feature and base surface.
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paper_title: Pasting Spline Surfaces

paper_content:

Details can be added to spline surfaces by applying displacement maps. However, the computational intensity of displacement mapping prevents its useful for interactive design. In this paper we explore a form of simulated displacement that can be used for interactive design by providing a preview of true displacements at low computational cost.
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paper_title: Variational Design and Fairing of Spline Surfaces

paper_content:

Variational principles have become quite popular in the design of free form surfaces. Among others they are used for fairing purposes. The choice of the ‘right’ fairness functional is a crucial step. There is always a tradeoff between high quality and computational effort. In this paper we present fairness functionals that allow fairing efficiently, i.e., produce high quality surfaces in a reasonable amount of time. These functionals can be considered as simplified thin plate energy functionals for parametric surfaces or as simplified MVC functionals.
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paper_title: Smoothing by spline functions. II

paper_content:

In this paper we generalize the results of [4] and modify the algorithm presented there to obtain a better rate of convergence.
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paper_title: Automatic smoothing with geometric surface patches

paper_content:

Abstract The generation of smooth surfaces from a mesh of three-dimensional data points is an important problem in the field of Computer Aided Geometric Design. A new smoothing method based upon generalized Coons' patches and calculus of variation is presented.
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paper_title: Fairing cubic B-spline curves

paper_content:

Abstract Algorithms are presented for locally fairing a B-spline curve in order to produce a pleasant curvature plot. The methods are based on inversions of the knot insertion algorithm for B-spline curves.
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paper_title: Preventing self-intersection under free-form deformation

paper_content:

Free-Form Deformation (FFD) is a versatile and efficient modeling technique which transforms an object by warping the surrounding space. The conventional user-interface is a lattice of movable control points but this tends to be cumbersome and counterintuitive. Directly Manipulated Free-Form Deformation (DMFFD) allows the user to drag object points directly and has proven useful in an interactive sculpting context. A serious shortcoming of both FFD and DMFFD is that some deformations cause self-intersection of the object. This is unrealistic and compromises the object's validity and suitability for later use. An in-built self-intersection test is thus required for FFD and its extensions to be truly robust In this paper, we present the following novel results set of theoretical conditions for preventing self-intersection by ensuring the injectivity (one-to-one mapping) of FFD, an exact. (necessary and sufficient) injectivity test which is accurate but computationally costly, an efficient but approximate injectivity test which is a sufficient condition only, and a new form of DMFFD which acts by composing many small injective deformations. The latter expands the range of possible deformations without sacrificing the speed of the approximate test.
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paper_title: Extended free-form deformation: a sculpturing tool for 3D geometric modeling

paper_content:

Current research efforts focus on providing more efficient and effective design methods for 3D modeling systems. In this paper a new deformation technique is presented. Among other things, arbitrarily shaped bumps can be designed and surfaces can be bent along arbitrarily shaped curves.The purpose of this research is to define a highly interactive and intuitive modeling technique for designers and stylists. A natural way of thinking is to mimic traditional trades, such as sculpturing and moulding.Furthermore, with this deformation technique, the modeling tool paradigm is introduced. The object is deformed with a user-defined deformation tool.This method is an extension of the Free-Form Deformation (FFD) technique proposed by Sederberg and Parry [17].
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paper_title: Solid modeling and beyond

paper_content:

A survey of the field of solid modeling and an assessment of its strengths and limitations are presented. The survey covers mathematical foundations, representations, algorithms, applications, user interfaces and systems. The primary conferences for each of these five aspects of solid modeling are listed. >
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paper_title: Solid Modeling: A Historical Summary and Contemporary Assessment

paper_content:

A new generation of industrial geometry systems is emerging based on the technology of the 1970's. The generations of the eighties and nineties will require more research.
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paper_title: Multiresolution Representation and Visualization of Volume Data

paper_content:

A system to represent and visualize scalar volume data at multiple resolution is presented. The system is built on a multiresolution model based on tetrahedral meshes with scattered vertices that can be obtained from any initial dataset. The model is built off-line through data simplification techniques, and stored in a compact data structure that supports fast on-line access. The system supports interactive visualization of a representation at an arbitrary level of resolution through isosurface and projective methods. The user can interactively adapt the quality of visualization to requirements of a specific application task and to the performance of a specific hardware platform. Representations at different resolutions can be used together to further enhance interaction and performance through progressive and multiresolution rendering.
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paper_title: Wavelets on a Bounded Interval

paper_content:

The aim of this paper is to present two different approaches to the study of multiresolution analysis and wavelets on a bounded interval. Recently, Meyer obtained orthonormal wavelets on a bounded interval by restricting Daubechies’ scaling functions and wavelets to [0, 1] and applying the Gram-Schmidt procedure to orthonormalize the restrictions. Our own approach — presented in the second part of the paper — is based on the semi-orthogonal Chui-Wang spline-wavelets. In this case we no longer have orthogonality in one scale, but there are explicit formulae for these wavelets.
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paper_title: Wavelets for computer graphics - theory and applications

paper_content:

1 Introduction 2 HAAR: The Simplest Wavelet Basis 3 Image Compression 4 Image Editing 5 Image Querying 6 Subdivision Curves 7 The Theory of Multiresolution Analysis 8 Multiresolution Curves 9 Multiresolution Tiling 10 Surface Wavelets 11 Surface Applications 12 Variational Modeling 13 Global Illumination
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paper_title: Multiresolution analysis for surfaces of arbitrary topological type

paper_content:

Multiresolution analysis and wavelets provide useful and efficient tools for representing functions at multiple levels of detail. Wavelet representations have been used in a broad range of applications, including image compression, physical simulation, and numerical analysis. In this article, we present a new class of wavelets, based on subdivision surfaces, that radically extends the class of representable functions. Whereas previous two-dimensional methods were restricted to functions difined on R 2 , the subdivision wavelets developed here may be applied to functions defined on compact surfaces of arbitrary topological type. We envision many applications of this work, including continuous level-of-detail control for graphics rendering, compression of geometric models, and acceleration of global illumination algorithms. Level-of-detail control for spherical domains is illustrated using two examples: shape approximation of a polyhedral model, and color approximation of global terrain data.
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paper_title: B.: Area preserving deformation of multiresolution curves

paper_content:

We describe a method for multiresolution deformation of closed planar curves that keeps the enclosed area constant. We use a wavelet based multiresolution representation of the curves that are represented by a finite number of control points at each level of resolution. A deformation can then be applied to the curve by modifying one or more control points at any level of resolution. This process is generally known as multiresolution editing to which we add the constraint of constant area. The key contribution of this paper is the efficient computation of the area in the wavelet decomposition form: the area is expressed through all levels of resolution as a bilinear form of the coarse and detail coefficients, and recursive formulas are developed to compute the matrix of this bilinear form. A similar result is also given for the bending energy of the curve. The area constraint is maintained through an optimization process. These contributions allow a real time multiresolution deformation with area constraint of complex curves.
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paper_title: Spline-Wavelets of Minimal Support

paper_content:

For any finite dimensional univariate spline spaces S 0 and S 1 of the same degree, with S 0 ⊂ S 1, we determine a basis for the orthogonal complement of S 0 in S 1. This basis has minimal support and is of interest in wavelet decompositions.
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paper_title: Animating soft substances with implicit surfaces

paper_content:

This paper presents a hybrid model for animation of soft inelastic substance which undergo topological changes, e.g. separation and fusion and which fit with the objects they are in contact with. The model uses a particle system coated with a smooth iso-surface that is used for performing collision detection, precise contact modeling and integration of response forces. The animation technique solves three problems inherent in implicit modeling. Firstly, local volume controllers are defined to insure constant volume deformation, even during highly inelastic processes such as splitting or fusion. Secondly, we avoid unwanted distance blending between disconnected pieces of the same substance. Finally, we simulate both collisions and progressive merging under compression between implicit surfaces that do not blend together. Parameter tuning is facilitated by the layered model and animation is generated at interactive rates.
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paper_title: Extending the CSG Tree. Warping, Blending and Boolean Operations in an Implicit Surface Modeling System

paper_content:

Automatic blending has characterized the major advantage of implicit surface modeling systems. Recently, the introduction of deformations based on space warping and Boolean operations between primitives has increased the usefulness of such systems. We propose a further enhancement which will extend the range of models that can be easily and intuitively defined with a skeletal implicit surface system. We describe a hierarchical method which allows arbitrary compositions of models that make use of blending, warping and Boolean operations. We call this structure the BlobTree. Blending and space warping are treated in the same way as union, difference and intersection, i.e. as nodes in the BlobTree. The traversal of the BlobTree is described along with two rendering algorithms ; a polygonizer and a ray tracer. We present some examples of interesting models which can be made easily using our approach that would be very difficult to represent with conventional systems.
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paper_title: Dynamic Catmull-Clark Subdivision Surfaces

paper_content:

Recursive subdivision schemes have been extensively used in computer graphics, computer-aided geometric design, and scientific visualization for modeling smooth surfaces of arbitrary topology. Recursive subdivision generates a visually pleasing smooth surface in the limit from an initial user-specified polygonal mesh through the repeated application of a fixed set of subdivision rules. We present a new dynamic surface model based on the Catmull-Clark subdivision scheme, a popular technique for modeling complicated objects of arbitrary genus. Our new dynamic surface model inherits the attractive properties of the Catmull-Clark subdivision scheme, as well as those of the physics-based models. This new model provides a direct and intuitive means of manipulating geometric shapes, and an efficient hierarchical approach for recovering complex shapes from large range and volume data sets using very few degrees of freedom (control vertices). We provide an analytic formulation and introduce the "physical" quantities required to develop the dynamic subdivision surface model which can be interactively deformed by applying synthesized forces. The governing dynamic differential equation is derived using Lagrangian mechanics and the finite element method. Our experiments demonstrate that this new dynamic model has a promising future in computer graphics, geometric shape design, and scientific visualization.
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paper_title: Dynamic NURBS with geometric constraints for interactive sculpting

paper_content:

This article develops a dynamic generalization of the nonuniform rational B-spline (NURBS) model. NURBS have become a defacto standard in commercial modeling systems because of their power to represent free-form shapes as well as common analytic shapes. To date, however, they have been viewed as purely geometric primitives that require the user to manually adjust multiple control points and associated weights in order to design shapes. Dynamic NURBS, or D-NURBS, are physics-based models that incorporate mass distributions, internal deformation energies, and other physical quantities into the popular NURBS geometric substrate. Using D-NURBS, a modeler can interactively sculpt curves and surfaces and design complex shapes to required specifications not only in the traditional indirect fashion, by adjusting control points and weights, but also through direct physical manipulation, by applying simulated forces and local and global shape constraints. D-NURBS move and deform in a physically intuitive manner in response to the user's direct manipulations. Their dynamic behavior results from the numerical integration of a set of nonlinear differential equations that automatically evolve the control points and weights in response to the applied forces and constraints. To derive these equations, we employ Lagrangian mechanics and a finite-element-like discretization. Our approach supports the trimming of D-NURBS surfaces using D-NURBS curves. We demonstrate D-NURBS models and constraints in applications including the rounding of solids, optimal surface fitting to unstructured data, surface design from cross sections, and free-form deformation. We also introduce a new technique for 2D shape metamorphosis using constrained D-NURBS surfaces.
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paper_title: Approximate Boolean operations on free-form solids

paper_content:

In this paper we describe a method for computing approximate results of boolcan operations (union, intersection, difference) applied to free-form solids bounded by multiresolution subdivision surfaces. We present algorithms for generating a control mesh for a multiresolution surface approximating the result, optimizing the parameterization of the new surface with respect to the original surfaces, and fitting the new surface to the geometry of the original surfaces. Our algorithms aim to minimize the size and optimize the quality of the new control mesh. The original control meshes are modified only in a neighborhood of the intersection. While the main goal is to obtain approximate results, high-accuracy approximations are also possible at additional computational expense, if the topology of the intersection curve is resolved correctly.
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paper_title: A multiresolution framework for variational subdivision

paper_content:

Subdivision is a powerful paradigm for the generaton of curves and surfaces. It is easy to implement, computationally efficient, and useful in a variety of applications because of its intimate connection with multiresolution analysis. An important task in computer graphics and geometric modeling is the construction of curves that interpolate a griven set of points and minimize a fairness functional (variational design). In the context of subdivision, fairing leads to special schemes requiring the solution of a banded linear system at every subdivision step. We present several examples of such schemes including one that reproduces nonuniform interpolating cubic splines. Expressing the construction in terms of certain elementary operations we are able to embed variational subdivision in the lifting framework, a powerful technique to construct wavelet filter banks given a subdivision scheme. This allows us to extend the traditional lifting scheme for FIR filters to a certain class of IIR filters. Consquently, we how how to build variationally optimal curves and associated, stable wavelets in a straightforward fashion. The algorithms to perform the corresponding decomposition and reconstruction transformations are easy to implement and efficient enough for interactive applications.
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paper_title: Subdivision Schemes for Thin Plate Splines

paper_content:

Thin plate splines are a well known entity of geometric design. They are defined as the minimizer of a variational problem whose differential operators approximate a simple notion of bending energy. Therefore, thin plate splines approximate surfaces with minimal bending energy and they are widely considered as the standard "fair" surface model. Such surfaces are desired for many modeling and design applications. ::: ::: ::: ::: Traditionally, the way to construct such surfaces is to solve the associated variational problem using finite elements or by using analytic solutions based on radial basis functions. This paper presents a novel approach for defining and computing thin plate splines using subdivision methods. We present two methods for the construction of thin plate splines based on subdivision: A globally supported subdivision scheme which exactly minimizes the energy functional as well as a family of strictly local subdivision schemes which only utilize a small, finite number of distinct subdivision rules and approximately solve the variational problem. A tradeoff between the accuracy of the approximation and the locality of the subdivision scheme is used to pick a particular member of this family of subdivision schemes. ::: ::: ::: ::: Later, we show applications of these approximating subdivision schemes to scattered data interpolation and the design of fair surfaces. In particular we suggest an efficient methodology for finding control points for the local subdivision scheme that will lead to an interpolating limit surface and demonstrate how the schemes can be used for the effective and efficient design of fair surfaces.
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paper_title: A variational approach to subdivision

paper_content:

Abstract In this paper a new class of interpolatory refinement schemes is presented which in every refinement step determine the new points by solving an optimization problem. In general, these schemes are global, i.e., every new point depends on all points of the polygon to be refined. By choosing appropriate quadratic functionals to be minimized iteratively during refinement, very efficient schemes producing limiting curves of high smoothness can be defined. The well known class of stationary interpolatory refinement schemes turns out to be a special case of these variational schemes.
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paper_title: Sharp features on multiresolution subdivision surfaces

paper_content:

In this paper we describe a method for creating sharp features and trim regions on multiresolution subdivision surfaces along a set of user-defined curves. Operations such as engraving, embossing, and trimming are important in many surface modeling applications. Their implementation, however, is non-trivial due to computational, topological, and smoothness constraints that the underlying surface has to satisfy. The novelty of our work lies in the ability to create sharp features anywhere on a surface and in the fact that the resulting representation remains within the multiresolution subdivision framework. Preserving the original representation has the advantage that other operations applicable to multiresolution subdivision surfaces can subsequently be applied to the edited model. We also introduce an extended set of subdivision rules for Catmull-Clark surfaces that allows the creation of creases along diagonals of control mesh faces.
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paper_title: The Lifting Scheme: A Construction Of Second Generation Wavelets

paper_content:

We present the lifting scheme, a simple construction of second generation wavelets; these are wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to a faster, in-place calculation of the wavelet transform. Several examples are included.
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paper_title: Dynamic Catmull-Clark Subdivision Surfaces

paper_content:

Recursive subdivision schemes have been extensively used in computer graphics, computer-aided geometric design, and scientific visualization for modeling smooth surfaces of arbitrary topology. Recursive subdivision generates a visually pleasing smooth surface in the limit from an initial user-specified polygonal mesh through the repeated application of a fixed set of subdivision rules. We present a new dynamic surface model based on the Catmull-Clark subdivision scheme, a popular technique for modeling complicated objects of arbitrary genus. Our new dynamic surface model inherits the attractive properties of the Catmull-Clark subdivision scheme, as well as those of the physics-based models. This new model provides a direct and intuitive means of manipulating geometric shapes, and an efficient hierarchical approach for recovering complex shapes from large range and volume data sets using very few degrees of freedom (control vertices). We provide an analytic formulation and introduce the "physical" quantities required to develop the dynamic subdivision surface model which can be interactively deformed by applying synthesized forces. The governing dynamic differential equation is derived using Lagrangian mechanics and the finite element method. Our experiments demonstrate that this new dynamic model has a promising future in computer graphics, geometric shape design, and scientific visualization.
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paper_title: Subdivision Methods for Geometric Design: A Constructive Approach

paper_content:

From the Publisher: ::: The world's leading animation houses rely increasingly on subdivision methods for creating realistic-looking complex shapes. However, until now there was no one book devoted to this powerful geometric modeling technique. Subdivision Methods for Geometric Design does the job with authority and precision, providing all that is needed to understand how subdivision works its magic, and how to make that magic work. ::: Throughout the book, icons cue readers to visit a companion Web site loaded with interactive exercises, implementations of the book's images, and supplementary material. Rich in theory, analysis, and practical information, this book is the complete resource for subdivision methods. ::: Features ::: The result of a collaboration between a leading university researcher and an industry practitioner. The only book devoted exclusively and comprehensively to this important new technology. Provides solid background and theoretical analysis of subdivision as well as a wide variety of specific applications. Addresses algorithms for Bezier and uniform B-Spline curves, Catmull-Clark subdivision for quad meshes, and regularity tests for polyhedral meshes. Via the companion Web site, (www.subdivision.com), provides opportunities for readers to experiment hands-on with implementations in a richly interactive environment. Includes a foreword by Tony DeRose, recipient of the 1999 ACM Computer Graphics Achievement Award for his seminal work in subdivision methods. ::: ::: Author Biography: Joe Warren, Professor of Computer Science at Rice University since 1986, is one of the world's leading experts on subdivision. Of his nearly 50 computer science papers-published in prestigious forums such as SIGGRAPH, Transactions on Graphics, Computer-Aided Geometric Design, and The Visual Computer-a dozen specifically address subdivision and its applications to computer graphics. Prof. Warren received both his M.S. and Ph.D. in Computer Science at Cornell University. His research interests focus on mathematical methods for representing geometric shape. ::: Henrik Weimer is a research scientist at the DaimlerChrysler Corporate Research Center in Berlin, where he works on knowledge-based support for the design and creation of engineering products. Dr. Weimer obtained his Ph.D. in Computer Science from Rice University.
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paper_title: Cut-and-paste editing of multiresolution surfaces

paper_content:

Cutting and pasting to combine different elements into a common structure are widely used operations that have been successfully adapted to many media types. Surface design could also benefit from the availability of a general, robust, and efficient cut-and-paste tool, especially during the initial stages of design when a large space of alternatives needs to be explored. Techniques to support cut-and-paste operations for surfaces have been proposed in the past, but have been of limited usefulness due to constraints on the type of shapes supported and the lack of real-time interaction. In this paper, we describe a set of algorithms based on multiresolution subdivision surfaces that perform at interactive rates and enable intuitive cut-and-paste operations.
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paper_title: A G1 triangular spline surface of arbitrary topological type

paper_content:

Abstract A piecewise G 1 spline surface composed of sextic triangular Bezier patches in one-to-one correspondence with the faces of a triangular control mesh is presented. Surfaces of arbitrary topological type are created by approximating any mesh that represents a triangulated 2-manifold. The surface has local support and is affine invariant. A set of shape parameters are available for additional local control over the shape of the surface. If some local regularities are present in the structure of the control mesh, then the corresponding patches may be parametrized as quintic or even quartic Bezier triangles. In the case of a regular triangulation (with appropriate shape parameters), the surface generated by this method is equivalent to a quartic C 2 triangular B-spline.
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paper_title: 3D sketching with fully free form deformation features (δ-F 4 ) for aesthetic design

paper_content:

This paper addresses the designers' activity and in particular the way designers express an object shape in 2D sketches through character lines. The tools currently available in commercial CAS/CAD systems to manipulate the digital models are still not sufficiently suited to support design. In this paper, we introduce the so-called Fully Free Form Deformation Features (δ-F4), able to take into account the curve-oriented stylists' way of working. Both the advantages of a free form surfaces deformation method and a feature-based approach are merged to define highlevel modelling entities allowing for a direct manipulation of surfaces through a restricted number of intuitive parameters. In addition, a δ-F4 classification is proposed to permit a fast access to the desired shape according to its semantics. The proposed approach is illustrated with some examples.
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paper_title: Animating soft substances with implicit surfaces

paper_content:

This paper presents a hybrid model for animation of soft inelastic substance which undergo topological changes, e.g. separation and fusion and which fit with the objects they are in contact with. The model uses a particle system coated with a smooth iso-surface that is used for performing collision detection, precise contact modeling and integration of response forces. The animation technique solves three problems inherent in implicit modeling. Firstly, local volume controllers are defined to insure constant volume deformation, even during highly inelastic processes such as splitting or fusion. Secondly, we avoid unwanted distance blending between disconnected pieces of the same substance. Finally, we simulate both collisions and progressive merging under compression between implicit surfaces that do not blend together. Parameter tuning is facilitated by the layered model and animation is generated at interactive rates.
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paper_title: Computing moments of objects enclosed by piecewise polynomial surfaces

paper_content:

Combining a polynomial free-form surface representation with Gauss' divergence theorem allows efficient and exact calculation of the moments of the enclosed objects. For example, for an cubic representation, volume, center of mass, and the inertia tensor can be computed in seconds even for complex objects with serval thousand patches while change due to local modification of the surface geometry can be computed in real-time as feedback for animation or design. Speed and simplicity of the approach allow solving the inverse problem of modeling to match prescribed moments.
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paper_title: B.: Area preserving deformation of multiresolution curves

paper_content:

We describe a method for multiresolution deformation of closed planar curves that keeps the enclosed area constant. We use a wavelet based multiresolution representation of the curves that are represented by a finite number of control points at each level of resolution. A deformation can then be applied to the curve by modifying one or more control points at any level of resolution. This process is generally known as multiresolution editing to which we add the constraint of constant area. The key contribution of this paper is the efficient computation of the area in the wavelet decomposition form: the area is expressed through all levels of resolution as a bilinear form of the coarse and detail coefficients, and recursive formulas are developed to compute the matrix of this bilinear form. A similar result is also given for the bending energy of the curve. The area constraint is maintained through an optimization process. These contributions allow a real time multiresolution deformation with area constraint of complex curves.
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paper_title: Principles of traditional animation applied to 3D computer animation

paper_content:

This paper describes the basic principles of traditional 2D hand drawn animation and their application to 3D computer animation. After describing how these principles evolved, the individual principles are detailed, addressing their meanings in 2D hand drawn animation and their application to 3D computer animation. This should demonstrate the importance of these principles to quality 3D computer animation.
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paper_title: 3D sketching with fully free form deformation features (δ-F 4 ) for aesthetic design

paper_content:

This paper addresses the designers' activity and in particular the way designers express an object shape in 2D sketches through character lines. The tools currently available in commercial CAS/CAD systems to manipulate the digital models are still not sufficiently suited to support design. In this paper, we introduce the so-called Fully Free Form Deformation Features (δ-F4), able to take into account the curve-oriented stylists' way of working. Both the advantages of a free form surfaces deformation method and a feature-based approach are merged to define highlevel modelling entities allowing for a direct manipulation of surfaces through a restricted number of intuitive parameters. In addition, a δ-F4 classification is proposed to permit a fast access to the desired shape according to its semantics. The proposed approach is illustrated with some examples.
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Title: Constraint Modeling for Curves and Surfaces in CAGD: A Survey

Section 1: Introduction
Description 1: Introduce the significance and advancements in curve and surface modeling in CAGD, highlighting the role and need for constraint-based modeling.

Section 2: Modeling Approaches in CAD
Description 2: Discuss various CAD modeling approaches including parametric and variational modeling, feature-based approaches, and declarative modeling, emphasizing the concept of constraint handling in each.

Section 3: Constraint-based Free-form Modeling Techniques
Description 3: Provide an overview of different techniques for constraint-based free-form modeling for curves, surfaces, and solids, particularly focusing on methods such as free-form parametric curves and surfaces, interactive modeling, variational modeling, feature-based modeling, multiresolution modeling, implicit surfaces, and subdivision surfaces.

Section 4: Classifying Constraints for Curve and Surface Modeling
Description 4: Propose a taxonomy of constraints used in curve and surface modeling, addressing different types of geometric and engineering constraints and their expression.

Section 5: Solving a Set of Constraints
Description 5: Discuss the process of solving constraints in free-form modeling, highlighting challenges such as under-constrained, iso-constrained, and over-constrained configurations.

Section 6: Open Issues
Description 6: Identify future research topics and open issues in constraint-based modeling in CAGD, focusing on product-independent and product-view-dependent constraints.

Section 7: Conclusion
Description 7: Summarize the survey's key points and emphasize the importance of continued research in constraint-based modeling for curve and surface design in CAGD.