A survey of partial differential equations in geometric design
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paper_title: Curves and surfaces for computer aided geometric design: A practical guide

paper_content:

From the Publisher: ::: This book will be of interest to computer graphics enthusiasts, software developers for CAD/CAM systems, geometric modeling researchers, graphics programmers, academicians, and many others throughout the graphics community. Assuming only a background in calculus and basic linear algebra, the author's informal and reader-friendly style makes the material accessible to a wide audience. Finally, the included disk contains data sets and all of the C programs used in the book, making it easier for the user to gain first-hand experience with the concepts as they are explained. This unified treatment of curve and surface design concepts focuses on Bezier and B-spline methods for curves, rational Bezier and B-spline curves, geometric continuity, spline interpolation, and Coons methods. The fourth edition has been thoroughly updated and revised to include a new chapter on recursive subdivision, as well as new sections on triangulations and scattered data interpolants. Finally, the disk in the back of the book has been updated to include all of the programs, as well as the data sets from the text.
---
paper_title: On harmonic and biharmonic Bézier surfaces ✩

paper_content:

We present a new method of surface generation from prescribed boundaries based on the elliptic partial differential operators. In particular, we focus on the study of the so-called harmonic and biharmonic Bezier surfaces. The main result we report here is that any biharmonic Bezier surface is fully determined by the boundary control points. We compare the new method, by way of practical examples, with some related methods such as surfaces generation using discretisation masks and functional minimisations.
---
paper_title: Curves and surfaces for computer aided geometric design: A practical guide

paper_content:

From the Publisher: ::: This book will be of interest to computer graphics enthusiasts, software developers for CAD/CAM systems, geometric modeling researchers, graphics programmers, academicians, and many others throughout the graphics community. Assuming only a background in calculus and basic linear algebra, the author's informal and reader-friendly style makes the material accessible to a wide audience. Finally, the included disk contains data sets and all of the C programs used in the book, making it easier for the user to gain first-hand experience with the concepts as they are explained. This unified treatment of curve and surface design concepts focuses on Bezier and B-spline methods for curves, rational Bezier and B-spline curves, geometric continuity, spline interpolation, and Coons methods. The fourth edition has been thoroughly updated and revised to include a new chapter on recursive subdivision, as well as new sections on triangulations and scattered data interpolants. Finally, the disk in the back of the book has been updated to include all of the programs, as well as the data sets from the text.
---
paper_title: Subdivision surfaces in character animation

paper_content:

The creation of believable and endearing characters in computer graphics presents a number of technical challenges, including the modeling, animation and rendering of complex shapes such as heads, hands, and clothing. Traditionally, these shapes have been modeled with NURBS surfaces despite the severe topological restrictions that NURBS impose. In order to move beyond these restrictions, we have recently introduced subdivision surfaces into our production environment. Subdivision surfaces are not new, but their use in high-end CG production has been limited. Here we describe a series of developments that were required in order for subdivision surfaces to meet the demands of high-end production. First, we devised a practical technique for constructing provably smooth variable-radius fillets and blends. Second, we developed methods for using subdivision surfaces in clothing simulation including a new algorithm for efficient collision detection. Third, we developed a method for constructing smooth scalar fields on subdivision surfaces, thereby enabling the use of a wider class of programmable shaders. These developments, which were used extensively in our recently completed short film Geri’s game, have become a highly valued feature of our production environment. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling; I.3.3 [Computer Graphics]: Picture/Image Generation.
---
paper_title: Curves and surfaces for computer aided geometric design: A practical guide

paper_content:

From the Publisher: ::: This book will be of interest to computer graphics enthusiasts, software developers for CAD/CAM systems, geometric modeling researchers, graphics programmers, academicians, and many others throughout the graphics community. Assuming only a background in calculus and basic linear algebra, the author's informal and reader-friendly style makes the material accessible to a wide audience. Finally, the included disk contains data sets and all of the C programs used in the book, making it easier for the user to gain first-hand experience with the concepts as they are explained. This unified treatment of curve and surface design concepts focuses on Bezier and B-spline methods for curves, rational Bezier and B-spline curves, geometric continuity, spline interpolation, and Coons methods. The fourth edition has been thoroughly updated and revised to include a new chapter on recursive subdivision, as well as new sections on triangulations and scattered data interpolants. Finally, the disk in the back of the book has been updated to include all of the programs, as well as the data sets from the text.
---
paper_title: Discrete surface modelling using partial differential equations

paper_content:

We use various nonlinear partial differential equations to efficiently solve several surface modelling problems, including surface blending, N-sided hole filling and free-form surface fitting. The nonlinear equations used include two second order flows, two fourth order flows and two sixth order flows. These nonlinear equations are discretized based on discrete differential geometry operators. The proposed approach is simple, efficient and gives very desirable results, for a range of surface models, possibly having sharp creases and corners.
---
paper_title: Discrete surface modelling using partial differential equations

paper_content:

We use various nonlinear partial differential equations to efficiently solve several surface modelling problems, including surface blending, N-sided hole filling and free-form surface fitting. The nonlinear equations used include two second order flows, two fourth order flows and two sixth order flows. These nonlinear equations are discretized based on discrete differential geometry operators. The proposed approach is simple, efficient and gives very desirable results, for a range of surface models, possibly having sharp creases and corners.
---
paper_title: A shape design system using volumetric implicit PDEs

paper_content:

Abstract Solid modeling based on partial differential equations (PDEs) can potentially unify both geometric constraints and functional requirements within a single design framework to model real-world objects via its explicit, direct integration with parametric geometry. In contrast, implicit functions indirectly define geometric objects as the level-set of underlying scalar fields. To maximize the modeling potential of PDE-based methodology, in this paper we tightly couple PDEs with volumetric implicit functions in order to achieve interactive, intuitive shape representation, manipulation, and deformation. In particular, the unified approach can reconstruct the PDE geometry of arbitrary topology from scattered data points or a set of sketch curves. We make use of elliptic PDEs for boundary value problems to define the volumetric implicit function. The proposed implicit PDE model has the capability to reconstruct a complete solid model from partial information and facilitates the direct manipulation of underlying volumetric datasets via sketch curves and iso-surface sculpting, deformation of arbitrary interior regions, as well as a set of CSG operations inside the working space. The prototype system that we have developed allows designers to interactively sketch the curve outlines of the object, define intensity values and gradient directions, and specify interpolatory points in the 3D working space. The governing implicit PDE treats these constraints as generalized boundary conditions to determine the unknown scalar intensity values over the entire working space. The implicit shape is reconstructed with specified intensity value accordingly and can be deformed using a set of sculpting toolkits. We use the finite-difference discretization and variational interpolating approach with the localized iterative solver for the numerical integration of our PDEs in order to accommodate the diversity of generalized boundary and additional constraints.
---
paper_title: Discrete surface modelling using partial differential equations

paper_content:

We use various nonlinear partial differential equations to efficiently solve several surface modelling problems, including surface blending, N-sided hole filling and free-form surface fitting. The nonlinear equations used include two second order flows, two fourth order flows and two sixth order flows. These nonlinear equations are discretized based on discrete differential geometry operators. The proposed approach is simple, efficient and gives very desirable results, for a range of surface models, possibly having sharp creases and corners.
---
paper_title: Image inpainting

paper_content:

Inpainting, the technique of modifying an image in an undetectable form, is as ancient as art itself. The goals and applications of inpainting are numerous, from the restoration of damaged paintings and photographs to the removal/replacement of selected objects. In this paper, we introduce a novel algorithm for digital inpainting of still images that attempts to replicate the basic techniques used by professional restorators. After the user selects the regions to be restored, the algorithm automatically fills-in these regions with information surrounding them. The fill-in is done in such a way that isophote lines arriving at the regions' boundaries are completed inside. In contrast with previous approaches, the technique here introduced does not require the user to specify where the novel information comes from. This is automatically done (and in a fast way), thereby allowing to simultaneously fill-in numerous regions containing completely different structures and surrounding backgrounds. In addition, no limitations are imposed on the topology of the region to be inpainted. Applications of this technique include the restoration of old photographs and damaged film; removal of superimposed text like dates, subtitles, or publicity; and the removal of entire objects from the image like microphones or wires in special effects.
---
paper_title: Variational problems and partial differential equations on implicit surfaces

paper_content:

A novel framework for solving variational problems and partial differential equations for scalar and vector-valued data defined on surfaces is introduced in this paper. The key idea is to implicitly represent the surface as the level set of a higher dimensional function and to solve the surface equations in a fixed Cartesian coordinate system using this new embedding function. The equations are then both intrinsic to the surface and defined in the embedding space. This approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces, as is commonly done in the literature. We describe the framework and present examples in computer graphics and image processing applications, including texture synthesis, flow field visualization, and image and vector field intrinsic regularization for data defined on 3D surfaces.
---
paper_title: Interactive shape design using volumetric implicit PDEs

paper_content:

Solid modeling based on Partial Differential Equations (PDEs) can potentially unify both geometric constraints and functional requirements within a single design framework to model real-world objects via its explicit, direct integration with parametric geometry. In contrast, implicit functions indirectly define geometric objects as the level-set of underlying scalar fields. To maximize the modeling potential of PDE-based methodology, in this paper we tightly couple PDEs with volumetric implicit functions in order to achieve interactive, intuitive shape representation, manipulation, and deformation. In particular, the unified approach can reconstruct the PDE geometry of arbitrary topology from scattered data points or a set of sketch curves. We make use of a fourth-order elliptic PDE to define the volumetric implicit function. The proposed implicit PDE model has the capability to reconstruct a complete solid model from partial information and facilitates the direct manipulation of underlying volumetric datasets via sketch curves, iso-surface sculpting, deformation of arbitrary interior regions, as well as a set of CSG operations inside the working space.The prototype system that we have developed allows designers to interactively sketch the curve outlines of the object, define intensity values and gradient directions, and specify interpolatory points in the 3D working space. The governing implicit PDE treats these constraints as generalized boundary conditions to determine the unknown scalar intensity values over the entire working space. The implicit shape is reconstructed with specified intensity value accordingly and can be deformed using a set of sculpting toolkits. We use the finite-difference discretization and variational interpolating approach with the localized iterative solver for the numerical integration of our PDEs in order to accommodate the diversity of generalized boundary constraints.
---
paper_title: Direct Manipulation and Interactive Sculpting of PDE Surfaces

paper_content:

This paper presents an integrated approach and a unified algorithm that combine the benefits of PDE surfaces and powerful physics-based modeling techniques within one single modeling framework, in order to realize the full potential of PDE surfaces. We have developed a novel system that allows direct manipulation and interactive sculpting of PDE surfaces at arbitrary location, hence supporting various interactive techniques beyond the conventional boundary control. Our prototype software affords users to interactively modify point, normal, curvature, and arbitrary region of PDE surfaces in a predictable way. We employ several simple, yet effective numerical techniques including the finite-difference discretization of the PDE surface, the multigrid-like subdivision on the PDE surface, the mass-spring approximation of the elastic PDE surface, etc. to achieve real-time performance. In addition, our dynamic PDE surfaces can also be approximated using standard bivariate B-spline finite elements, which can subsequently be sculpted and deformed directly in real-time subject to intrinsic PDE constraints. Our experiments demonstrate many attractive advantages of our dynamic PDE formulation such as intuitive control, real-time feedback, and usability to the general public.
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paper_title: Techniques for interactive design using the PDE method

paper_content:

Interactive design of practical surfaces using the partial differential equation (PDE) method is considered. The PDE method treats surface design as a boundary value problem (ensuring that surfaces can be defined using a small set of design parameters). Owing to the elliptic nature of the PDE operator, the boundary conditions imposed around the edges of the surface control the internal shape of the surface. Moreover, surfaces obtained in this manner tend to be smooth and fair. The PDE chosen has a closed form solution allowing the interactive manipulation of the surfaces in real time. Thus we present efficient techniques by which we show how surfaces of practical significance can be constructed interactively in real time.
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paper_title: Parametric Design and Optimisation of Thin-Walled Structures for Food Packaging

paper_content:

In this paper the parametric design and functional optimisation of thin-walled structures made from plastics for food packaging is considered. These objects are produced in such vast numbers each year that one important task in the design of these objects is to minimise the amount of plastic used, subject to functional constraints, to reduce the costs of production and to conserve raw materials. By means of performing an automated optimisation on the possible shapes of the food containers, where the geometry is parametrised succinctly, a strategy to create the optimal design of the containers subject to a given set of functional constraints is demonstrated.
---
paper_title: Generating blend surfaces using partial differential equations

paper_content:

Abstract A method is proposed for representing surfaces as solutions to partial differential equations. It is shown, by examples from the field of blend generation, that the method can easily achieve the required degree of continuity between the blend and the surfaces to which it attaches. The surfaces also have the property of being geometrically ‘well-behaved’.
---
paper_title: On harmonic and biharmonic Bézier surfaces ✩

paper_content:

We present a new method of surface generation from prescribed boundaries based on the elliptic partial differential operators. In particular, we focus on the study of the so-called harmonic and biharmonic Bezier surfaces. The main result we report here is that any biharmonic Bezier surface is fully determined by the boundary control points. We compare the new method, by way of practical examples, with some related methods such as surfaces generation using discretisation masks and functional minimisations.
---
paper_title: Spectral approximations to PDE surfaces

paper_content:

The PDE method generates surfaces from the solutions to elliptic partial differential equations (PDEs), where boundary conditions are used to control surface shape. This paper describes a method whereby PDE surfaces may be obtained in closed form, even for the case of general boundary conditions. Furthermore, the method is fast, making possible the interactive manipulations of PDE surfaces in real-time.
---
paper_title: Using partial differential equations to generate free-form surfaces

paper_content:

Abstract A method of generating free-form surfaces using solutions to a suitably chosen partial differential equation is discussed. By varying the boundary conditions and a parameter in the partial differential equation, it is demonstrated that a wide variety of surface shapes are accessible to the method.
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paper_title: An analytic pseudo-spectral method to generate a regular 4-sided PDE surface patch

paper_content:

We describe a pseudo-spectral method for rapidly calculating an analytic approximation to a 4-sided PDE surface patch. The method generates an approximate solution consisting of three parts: an eigenfunction solution and a polynomial solution, both of which satisfy the generating partial differential equation exactly, and a third function, or 'remainder' term that ensures that the boundary conditions are satisfied exactly. Being analytic, the approximation allows an arbitrary degree of surface refinement thereby facilitating physical analysis.
---
paper_title: Generating blend surfaces using partial differential equations

paper_content:

Abstract A method is proposed for representing surfaces as solutions to partial differential equations. It is shown, by examples from the field of blend generation, that the method can easily achieve the required degree of continuity between the blend and the surfaces to which it attaches. The surfaces also have the property of being geometrically ‘well-behaved’.
---
paper_title: Variational geometry in computer-aided design

paper_content:

A system has been developed which utilizes variational geometry in the design and modification of mechanical parts. Three-dimensional constraints between characteristic points are used to define an object's geometry. Modification of geometry is accomplished by alteration of one or more constraints. A matrix method is used to determine the shape of the part by simultaneous solution of constraint equations. A method for increasing the speed and efficiency of the solution procedure is described. The method uses the relationships between the geometry and constraints to minimize the number of equations and variables to be solved.
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paper_title: Spectral approximations to PDE surfaces

paper_content:

The PDE method generates surfaces from the solutions to elliptic partial differential equations (PDEs), where boundary conditions are used to control surface shape. This paper describes a method whereby PDE surfaces may be obtained in closed form, even for the case of general boundary conditions. Furthermore, the method is fast, making possible the interactive manipulations of PDE surfaces in real-time.
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paper_title: Fast Surface Modelling Using a 6th Order PDE

paper_content:

Although the control-point based parametric approach is used most widely in free-form surface modelling, complementary techniques co-exist to meet various specialised requirements. The partial differential equation (PDE) based modelling approach is especially suitable for satisfying surface boundary constraints. They are also effective for the generation of families of free-form surfaces, which share a common base and differ in their secondary features. In this paper, we present a fast surface modelling method using a sixth order PDE. This PDE provides enough degrees of freedom not only to accommodate tangent, but also curvature boundary conditions and offers more shape control parameters to serve as user controls for the manipulation of surface shapes. In order to achieve real-time performance, we have constructed a surface function and developed a high-precision approximate solution to the 6th order PDE. Unlike some existing PDE-based techniques, this resolution method can satisfy the boundary conditions exactly, and is able to create free-form surfaces as fast and almost as accurately as the closed-form (analytical) solutions. Due to the fact that it has sufficient degrees of freedom to accommodate the continuity of 3-sided and 4-sided surface patches at their boundaries, this method is able to model complex surfaces consisting of multiple patches. Compared with existing PDE-based modelling methods, this method is both fast and can solve a larger class of surface modelling problems. ::: ::: Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Curves, surfaces, solid, and object representations; physically based modelling
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paper_title: Dynamic PDE-based surface design using geometric and physical constraints

paper_content:

PDE surfaces, which are defined as solutions of partial differential equations (PDEs), offer many modeling advantages in surface blending, free-form surface modeling, and specifying surface's aesthetic or functional requirements. Despite the earlier advances of PDE surfaces, previous PDE-based techniques exhibit certain difficulties such as lack of interactive sculpting capabilities and restrained topological structure of modeled objects. This paper presents an integrated approach that can incorporate PDE surfaces into the powerful physics-based modeling framework, to realize the full potential of PDE methodology. We have developed a prototype system that allows interactive design of flexible topological surfaces as PDE surfaces and displacements using generalized boundary conditions as well as a variety of geometric and physical constraints, hence supporting various interactive techniques beyond the conventional boundary control. The system offers a set of sculpting toolkits that allow users to interactively modify arbitrary points, curve spans, and/or regions of interest across the entire PDE surfaces and displacements in an intuitive and physically meaningful way. To achieve real-time performance, we employ several simple, yet efficient numerical techniques, including the finite-difference discretization, the multigrid-like subdivision, and the mass-spring approximation of elastic PDE surfaces and displacements. In addition, we present the standard bivariant B-spline finite element approximations of dynamic PDEs, which can subsequently be sculpted and deformed directly in real-time subject to the intrinsic PDE constraints. Our experiments demonstrate many attractive advantages of the physics-based PDE formulation such as intuitive control, real-time feedback, and usability to both professional and common users.
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paper_title: D-NURBS: A Physics-Based Framework for Geometric Design

paper_content:

Presents dynamic non-uniform rational B-splines (D-NURBS), a physics-based generalization of NURBS. NURBS have become a de facto standard in commercial modeling systems. Traditionally, however, NURBS have been viewed as purely geometric primitives, which require the designer to interactively adjust many degrees of freedom-control points and associated weights-to achieve the desired shapes. The conventional shape modification process can often be clumsy and laborious. D-NURBS are physics-based models that incorporate physical quantities into the NURBS geometric substrate. Their dynamic behavior, resulting from the numerical integration of a set of nonlinear differential equations, produces physically meaningful, and hence intuitive shape variation. Consequently, a modeler can interactively sculpt complex shapes to required specifications not only in the traditional indirect fashion, by adjusting control points and setting weights, but also through direct physical manipulation, by applying simulated forces and local and global shape constraints. We use Lagrangian mechanics to formulate the equations of motion for D-NURBS curves, tensor-product D-NURBS surfaces, swung D-NURBS surfaces and triangular D-NURBS surfaces. We apply finite element analysis to reduce these equations to efficient numerical algorithms computable at interactive rates on common graphics workstations. We implement a prototype modeling environment based on D-NURBS and demonstrate that D-NURBS can be effective tools in a wide range of computer-aided geometric design (CAGD) applications.
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paper_title: Integrating physics-based modeling with PDE solids for geometric design

paper_content:

PDE techniques, which use partial differential equations (PDEs) to model the shapes of various real-world objects, can unify their geometric attributes and functional constraints in geometric computing and graphics. This paper presents a unified dynamic approach that allows modelers to define the solid geometry of sculptured objects using the second-order or fourth-order elliptic PDEs subject to flexible boundary conditions. Founded upon the previous work on PDE solids by Bloor and Wilson (1989, 1990, 1993), as well as our recent research on the interactive sculpting of physics-based PDE surfaces, our new formulation and its associated dynamic principle permit designers to directly deform PDE solids whose behaviors are natural and intuitive subject to imposed constraints. Users can easily model and interact with solids of complicated geometry and/or arbitrary topology from locally-defined PDE primitives through trimming operations. We employ the finite-difference discretization and the multi-grid subdivision to solve the PDEs numerically. Our PDE-based modeling software offers users various sculpting toolkits for solid design, allowing them to interactively modify the physical and geometric properties of arbitrary points, curve spans, regions of interest (either in the isoparametric or nonisoparametric form) on boundary surfaces, as well as any interior parts of modeled objects.
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paper_title: Dynamic PDE surfaces with flexible and general geometric constraints

paper_content:

PDE surfaces, whose behavior is governed by partial differential equations (PDEs), have demonstrated many modeling advantages in surface blending, free-form surface modeling, and surface aesthetic or functional specifications. Although PDE surfaces can potentially unify geometric attributes and functional constraints for surface design, current PDE based techniques exhibit certain difficulties such as the restrained topological structure of modeled objects and the lack of interactive editing functionalities. We propose an integrated approach and develop a set of algorithms that augment conventional PDE surfaces with material properties and dynamic behavior. The authors incorporate PDE surfaces into the powerful physics based framework, aiming to realize the full potential of the PDE methodology. We have implemented a prototype software environment that can offer users a wide array of PDE surfaces with flexible topology (through trimming and joining operations) as well as generalized boundary constraints. Using our system, designers can dynamically manipulate PDE surfaces at arbitrary location with applied forces. Our sculpting toolkits allow users to interactively modify arbitrary point, curve span, and/or region of interest throughout the entire PDE surface in an intuitive and predictable way. To achieve real time sculpting, we employ several simple, yet efficient numerical techniques such as finite difference discretization, multi-grid subdivision, and FEM approximation. Our experiments demonstrate many advantages of physics based PDE formulation such as intuitive control, real time feedback, and usability to both professional and non-expert users.
---
paper_title: Using partial differential equations to generate free-form surfaces

paper_content:

Abstract A method of generating free-form surfaces using solutions to a suitably chosen partial differential equation is discussed. By varying the boundary conditions and a parameter in the partial differential equation, it is demonstrated that a wide variety of surface shapes are accessible to the method.
---
paper_title: Interactive shape design using volumetric implicit PDEs

paper_content:

Solid modeling based on Partial Differential Equations (PDEs) can potentially unify both geometric constraints and functional requirements within a single design framework to model real-world objects via its explicit, direct integration with parametric geometry. In contrast, implicit functions indirectly define geometric objects as the level-set of underlying scalar fields. To maximize the modeling potential of PDE-based methodology, in this paper we tightly couple PDEs with volumetric implicit functions in order to achieve interactive, intuitive shape representation, manipulation, and deformation. In particular, the unified approach can reconstruct the PDE geometry of arbitrary topology from scattered data points or a set of sketch curves. We make use of a fourth-order elliptic PDE to define the volumetric implicit function. The proposed implicit PDE model has the capability to reconstruct a complete solid model from partial information and facilitates the direct manipulation of underlying volumetric datasets via sketch curves, iso-surface sculpting, deformation of arbitrary interior regions, as well as a set of CSG operations inside the working space.The prototype system that we have developed allows designers to interactively sketch the curve outlines of the object, define intensity values and gradient directions, and specify interpolatory points in the 3D working space. The governing implicit PDE treats these constraints as generalized boundary conditions to determine the unknown scalar intensity values over the entire working space. The implicit shape is reconstructed with specified intensity value accordingly and can be deformed using a set of sculpting toolkits. We use the finite-difference discretization and variational interpolating approach with the localized iterative solver for the numerical integration of our PDEs in order to accommodate the diversity of generalized boundary constraints.
---
paper_title: Interactive design using higher order PDEs

paper_content:

This paper extends the PDE method of surface generation. The governing partial differential equation is generalised to sixth order to increase its flexibility. The PDE is solved analytically, even in the case of general boundary conditions, making the method fast. The boundary conditions, which control the surface shape, are specified interactively, allowing intuitive manipulation of generic shapes. A compact user interface is presented which makes use of direct manipulation and other techniques for 3D interaction.
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paper_title: Functionality in blend design

paper_content:

A method is presented for obtaining surfaces that satisfy certain given design conditions. The surfaces are generated as solutions to a partial differential equation and designs, which optimize some function of the surface while satisfying some other constraints, are found. Examples of maximizing the heat lost from a surface and of minimizing shear stress are considered.
---
paper_title: Discrete surface modelling using partial differential equations

paper_content:

We use various nonlinear partial differential equations to efficiently solve several surface modelling problems, including surface blending, N-sided hole filling and free-form surface fitting. The nonlinear equations used include two second order flows, two fourth order flows and two sixth order flows. These nonlinear equations are discretized based on discrete differential geometry operators. The proposed approach is simple, efficient and gives very desirable results, for a range of surface models, possibly having sharp creases and corners.
---
paper_title: Generating blend surfaces using partial differential equations

paper_content:

Abstract A method is proposed for representing surfaces as solutions to partial differential equations. It is shown, by examples from the field of blend generation, that the method can easily achieve the required degree of continuity between the blend and the surfaces to which it attaches. The surfaces also have the property of being geometrically ‘well-behaved’.
---
paper_title: Image inpainting

paper_content:

Inpainting, the technique of modifying an image in an undetectable form, is as ancient as art itself. The goals and applications of inpainting are numerous, from the restoration of damaged paintings and photographs to the removal/replacement of selected objects. In this paper, we introduce a novel algorithm for digital inpainting of still images that attempts to replicate the basic techniques used by professional restorators. After the user selects the regions to be restored, the algorithm automatically fills-in these regions with information surrounding them. The fill-in is done in such a way that isophote lines arriving at the regions' boundaries are completed inside. In contrast with previous approaches, the technique here introduced does not require the user to specify where the novel information comes from. This is automatically done (and in a fast way), thereby allowing to simultaneously fill-in numerous regions containing completely different structures and surrounding backgrounds. In addition, no limitations are imposed on the topology of the region to be inpainted. Applications of this technique include the restoration of old photographs and damaged film; removal of superimposed text like dates, subtitles, or publicity; and the removal of entire objects from the image like microphones or wires in special effects.
---
paper_title: Anisotropic diffusion of surfaces and functions on surfaces

paper_content:

We present a unified anisotropic geometric diffusion PDE model for smoothing (fairing) out noise both in triangulated two-manifold surface meshes in IR3 and functions defined on these surface meshes, while enhancing curve features on both by careful choice of an anisotropic diffusion tensor. We combine the C1 limit representation of Loop's subdivision for triangular surface meshes and vector functions on the surface mesh with the established diffusion model to arrive at a discretized version of the diffusion problem in the spatial direction. The time direction discretization then leads to a sparse linear system of equations. Iteratively solving the sparse linear system yields a sequence of faired (smoothed) meshes as well as faired functions.
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paper_title: Solving Variational Problems and Partial Differential Equations Mapping into General Target Manifolds

paper_content:

A framework for solving variational problems and partial differential equations that define maps onto a given generic manifold is introduced in this paper. We discuss the framework for arbitrary target manifolds, while the domain manifold problem was addressed in [J. Comput. Phys. 174(2) (2001) 759]. The key idea is to implicitly represent the target manifold as the level-set of a higher dimensional function, and then implement the equations in the Cartesian coordinate system where this embedding function is defined. In the case of variational problems, we restrict the search of the minimizing map to the class of maps whose target is the level-set of interest. In the case of partial differential equations, we re-write all the equation's geometric characteristics with respect to the embedding function. We then obtain a set of equations that, while defined on the whole Euclidean space, are intrinsic to the implicitly defined target manifold and map into it. This permits the use of classical numerical techniques in Cartesian grids, regardless of the geometry of the target manifold. The extension to open surfaces and submanifolds is addressed in this paper as well. In the latter case, the submanifold is defined as the intersection of two higher dimensional hypersurfaces, and all the computations are restricted to this intersection. Examples of the applications of the framework here described include harmonic maps in liquid crystals, where the target manifold is a hypersphere; probability maps, where the target manifold is a hyperplane; chroma enhancement; texture mapping; and general geometric mapping between high dimensional manifolds.
---
paper_title: Variational problems and partial differential equations on implicit surfaces

paper_content:

A novel framework for solving variational problems and partial differential equations for scalar and vector-valued data defined on surfaces is introduced in this paper. The key idea is to implicitly represent the surface as the level set of a higher dimensional function and to solve the surface equations in a fixed Cartesian coordinate system using this new embedding function. The equations are then both intrinsic to the surface and defined in the embedding space. This approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces, as is commonly done in the literature. We describe the framework and present examples in computer graphics and image processing applications, including texture synthesis, flow field visualization, and image and vector field intrinsic regularization for data defined on 3D surfaces.
---
paper_title: Discrete surface modelling using partial differential equations

paper_content:

We use various nonlinear partial differential equations to efficiently solve several surface modelling problems, including surface blending, N-sided hole filling and free-form surface fitting. The nonlinear equations used include two second order flows, two fourth order flows and two sixth order flows. These nonlinear equations are discretized based on discrete differential geometry operators. The proposed approach is simple, efficient and gives very desirable results, for a range of surface models, possibly having sharp creases and corners.
---
paper_title: Geometric Fairing of Irregular Meshes for Free-Form Surface Design

paper_content:

In this paper we present a new algorithm for smoothing arbitrary triangle meshes while satisfying G^1 boundary conditions. The algorithm is based on solving a nonlinear fourth order partial differential equation (PDE) that only depends on intrinsic surface properties instead of being derived from a particular surface parameterization. This continuous PDE has a (representation-independent) well-defined solution which we approximate by our triangle mesh. Hence, changing the mesh complexity (refinement) or the mesh connectivity (remeshing) leads to just another discretization of the same smooth surface and doesn't affect the resulting geometric shape beyond this. This is typically not true for filter-based mesh smoothing algorithms. To simplify the computation we factorize the fourth order PDE into a set of two nested second order problems thus avoiding the estimation of higher order derivatives. Further acceleration is achieved by applying multigrid techniques on a fine-to-coarse hierarchical mesh representation.
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paper_title: Efficient parametrization of generic aircraft geometry

paper_content:

A new method is presented for the parametrization of aircraft geometry; it is efficient in the sense that a relatively small number of "design" parameters are required to describe a complex surface geometry. The method views surface generation as a boundary-value problem and produces surfaces as the solutions to elliptic partial differential equations, hence its name, the PDE method. The use of the PDE method will be illustrated in this article by the parametrization of double delta geometries; it will be shown that it is possible to capture the basic features of the large-scale geometry of the aircraft in terms of a small set of design variables.
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paper_title: Parametric Design and Optimisation of Thin-Walled Structures for Food Packaging

paper_content:

In this paper the parametric design and functional optimisation of thin-walled structures made from plastics for food packaging is considered. These objects are produced in such vast numbers each year that one important task in the design of these objects is to minimise the amount of plastic used, subject to functional constraints, to reduce the costs of production and to conserve raw materials. By means of performing an automated optimisation on the possible shapes of the food containers, where the geometry is parametrised succinctly, a strategy to create the optimal design of the containers subject to a given set of functional constraints is demonstrated.
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paper_title: Efficient Shape Parametrisation for Automatic Design Optimisation using a Partial Differential Equation Formulation

paper_content:

Abstract This paper presents a methodology for efficient shape parametrisation for automatic design optimisation using a partial differential equation (PDE) formulation. It is shown how the choice of an elliptic PDE enables one to define and parametrise geometries corresponding to complex shapes. By using the PDE formulation it is shown how the shape definition and parametrisation can be based on a boundary value approach by which complex shapes can be created and parametrised based on the shape information at the boundaries or the character lines defining the shape. Furthermore, this approach to shape definition allows complex shapes to be parametrised intuitively using a very small set of design parameters. Thus, it is shown that the PDE based approach to shape parametrisation when combined with a standard method for numerical optimisation is capable of setting up automatic design optimisation problems allowing practical design optimisation to be more feasible.
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paper_title: Efficient parametrization of generic aircraft geometry

paper_content:

A new method is presented for the parametrization of aircraft geometry; it is efficient in the sense that a relatively small number of "design" parameters are required to describe a complex surface geometry. The method views surface generation as a boundary-value problem and produces surfaces as the solutions to elliptic partial differential equations, hence its name, the PDE method. The use of the PDE method will be illustrated in this article by the parametrization of double delta geometries; it will be shown that it is possible to capture the basic features of the large-scale geometry of the aircraft in terms of a small set of design variables.
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paper_title: Parametric Design and Optimisation of Thin-Walled Structures for Food Packaging

paper_content:

In this paper the parametric design and functional optimisation of thin-walled structures made from plastics for food packaging is considered. These objects are produced in such vast numbers each year that one important task in the design of these objects is to minimise the amount of plastic used, subject to functional constraints, to reduce the costs of production and to conserve raw materials. By means of performing an automated optimisation on the possible shapes of the food containers, where the geometry is parametrised succinctly, a strategy to create the optimal design of the containers subject to a given set of functional constraints is demonstrated.
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paper_title: Efficient Shape Parametrisation for Automatic Design Optimisation using a Partial Differential Equation Formulation

paper_content:

Abstract This paper presents a methodology for efficient shape parametrisation for automatic design optimisation using a partial differential equation (PDE) formulation. It is shown how the choice of an elliptic PDE enables one to define and parametrise geometries corresponding to complex shapes. By using the PDE formulation it is shown how the shape definition and parametrisation can be based on a boundary value approach by which complex shapes can be created and parametrised based on the shape information at the boundaries or the character lines defining the shape. Furthermore, this approach to shape definition allows complex shapes to be parametrised intuitively using a very small set of design parameters. Thus, it is shown that the PDE based approach to shape parametrisation when combined with a standard method for numerical optimisation is capable of setting up automatic design optimisation problems allowing practical design optimisation to be more feasible.
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paper_title: Subdivision surfaces in character animation

paper_content:

The creation of believable and endearing characters in computer graphics presents a number of technical challenges, including the modeling, animation and rendering of complex shapes such as heads, hands, and clothing. Traditionally, these shapes have been modeled with NURBS surfaces despite the severe topological restrictions that NURBS impose. In order to move beyond these restrictions, we have recently introduced subdivision surfaces into our production environment. Subdivision surfaces are not new, but their use in high-end CG production has been limited. Here we describe a series of developments that were required in order for subdivision surfaces to meet the demands of high-end production. First, we devised a practical technique for constructing provably smooth variable-radius fillets and blends. Second, we developed methods for using subdivision surfaces in clothing simulation including a new algorithm for efficient collision detection. Third, we developed a method for constructing smooth scalar fields on subdivision surfaces, thereby enabling the use of a wider class of programmable shaders. These developments, which were used extensively in our recently completed short film Geri’s game, have become a highly valued feature of our production environment. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling; I.3.3 [Computer Graphics]: Picture/Image Generation.
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paper_title: Discrete surface modelling using partial differential equations

paper_content:

We use various nonlinear partial differential equations to efficiently solve several surface modelling problems, including surface blending, N-sided hole filling and free-form surface fitting. The nonlinear equations used include two second order flows, two fourth order flows and two sixth order flows. These nonlinear equations are discretized based on discrete differential geometry operators. The proposed approach is simple, efficient and gives very desirable results, for a range of surface models, possibly having sharp creases and corners.
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paper_title: On the Spine of a PDE Surface

paper_content:

The spine of an object is an entity that can characterise the object's topology and describes the object by a lower dimension. It has an intuitive appeal for supporting geometric modelling operations. The aim of this paper is to show how a spine for a PDE surface can be generated. For the purpose of the work presented here an analytic solution form for the chosen PDE is utilised. It is shown that the spine of the PDE surface is then computed as a by-product of this analytic solution. This paper also discusses how the of a PDE surface can be used to manipulate the shape. The solution technique adopted here caters for periodic surfaces with general boundary conditions allowing the possibility of the spine based shape manipulation for a wide variety of free-form PDE surface shapes.
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paper_title: Subdivision surfaces in character animation

paper_content:

The creation of believable and endearing characters in computer graphics presents a number of technical challenges, including the modeling, animation and rendering of complex shapes such as heads, hands, and clothing. Traditionally, these shapes have been modeled with NURBS surfaces despite the severe topological restrictions that NURBS impose. In order to move beyond these restrictions, we have recently introduced subdivision surfaces into our production environment. Subdivision surfaces are not new, but their use in high-end CG production has been limited. Here we describe a series of developments that were required in order for subdivision surfaces to meet the demands of high-end production. First, we devised a practical technique for constructing provably smooth variable-radius fillets and blends. Second, we developed methods for using subdivision surfaces in clothing simulation including a new algorithm for efficient collision detection. Third, we developed a method for constructing smooth scalar fields on subdivision surfaces, thereby enabling the use of a wider class of programmable shaders. These developments, which were used extensively in our recently completed short film Geri’s game, have become a highly valued feature of our production environment. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling; I.3.3 [Computer Graphics]: Picture/Image Generation.
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paper_title: Animation and rendering of complex water surfaces

paper_content:

We present a new method for the animation and rendering of photo-realistic water effects. Our method is designed to produce visually plausible three dimensional effects, for example the pouring of water into a glass (see figure 1) and the breaking of an ocean wave, in a manner which can be used in a computer animation environment. In order to better obtain photorealism in the behavior of the simulated water surface, we introduce a new "thickened" front tracking technique to accurately represent the water surface and a new velocity extrapolation method to move the surface in a smooth, water-like manner. The velocity extrapolation method allows us to provide a degree of control to the surface motion, e.g. to generate a windblown look or to force the water to settle quickly. To ensure that the photorealism of the simulation carries over to the final images, we have integrated our method with an advanced physically based rendering system.
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paper_title: Visual simulation of smoke

paper_content:

In this paper, we propose a new approach to numerical smoke simulation for computer graphics applications. The method proposed here exploits physics unique to smoke in order to design a numerical method that is both fast and efficient on the relatively coarse grids traditionally used in computer graphics applications (as compared to the much finer grids used in the computational fluid dynamics literature). We use the inviscid Euler equations in our model, since they are usually more appropriate for gas modeling and less computationally intensive than the viscous Navier-Stokes equations used by others. In addition, we introduce a physically consistent vorticity confinement term to model the small scale rolling features characteristic of smoke that are absent on most coarse grid simulations. Our model also correctly handles the inter-action of smoke with moving objects.
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paper_title: Physically based modeling and animation of fire

paper_content:

We present a physically based method for modeling and animating fire. Our method is suitable for both smooth (laminar) and turbulent flames, and it can be used to animate the burning of either solid or gas fuels. We use the incompressible Navier-Stokes equations to independently model both vaporized fuel and hot gaseous products. We develop a physically based model for the expansion that takes place when a vaporized fuel reacts to form hot gaseous products, and a related model for the similar expansion that takes place when a solid fuel is vaporized into a gaseous state. The hot gaseous products, smoke and soot rise under the influence of buoyancy and are rendered using a blackbody radiation model. We also model and render the blue core that results from radicals in the chemical reaction zone where fuel is converted into products. Our method allows the fire and smoke to interact with objects, and flammable objects can catch on fire.
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Title: A Survey of Partial Differential Equations in Geometric Design

Section 1: Introduction
Description 1: This section introduces the historical and modern context of geometric design and outlines the evolution and usage of surface generation techniques, focusing on the role of PDEs.

Section 2: Common Surface Generation Techniques for Geometric Design
Description 2: This section describes the different methods used for surface generation in geometric design, such as B-splines, Bézier surfaces, NURBS, and other parametric surfaces.

Section 3: Partial Differential Equations
Description 3: This section defines what partial differential equations (PDEs) are and provides a general overview of their importance and application in surface generation.

Section 4: Geometric PDE Surfaces
Description 4: This section discusses the generation and advantages of PDE surfaces, emphasizing the types of PDEs used and their respective benefits.

Section 5: Implicit PDE Surfaces
Description 5: This section explains implicit PDE surfaces, describing various geometric flows and boundary conditions, and highlights studies relevant to computer-aided geometric design.

Section 6: Accounting for the Use of Elliptic PDEs to Obtain Implicit PDE Surfaces
Description 6: This section details the use of elliptic PDEs in generating implicit surfaces and discusses boundary conditions, numerical methods, and practical applications.

Section 7: Parametric PDE Surfaces
Description 7: This section covers parametric PDE surfaces, their advantages, and methods for their generation, including a detailed look at the Bloor-Wilson PDE method.

Section 8: Alternatives to the Bloor-Wilson PDE Method
Description 8: This section presents variations and alternatives to the Bloor-Wilson PDE method, including alternative elliptic PDE formulations and physics-based models.

Section 9: Applications of PDE Surfaces
Description 9: This section provides an overview of applications of PDE surfaces in computer-aided geometric design, including interactive design, shape blending, surface processing, and design analysis and optimization.

Section 10: Surface Generation
Description 10: This section discusses PDE-based surface generation techniques for interactive design and blending, detailing the methods and advantages.

Section 11: Surface Processing
Description 11: This section discusses processing existing surfaces using PDEs, with applications such as image inpainting, noise reduction, N-sided hole filling, and surface fairing.

Section 12: Design Analysis and Optimization
Description 12: This section explains how PDE surfaces can be used for design analysis and optimization by characterizing and optimizing object designs based on physical properties.

Section 13: Other Applications
Description 13: This section discusses additional applications of PDE surfaces in subdivision, geometric manipulations, and animation.

Section 14: Other Aspects of Computer Graphics Related to PDEs
Description 14: This section touches on the application of PDEs in simulating natural phenomena in computer graphics, such as water, smoke, and fire.

Section 15: Conclusions
Description 15: This section provides concluding remarks, summarizing the benefits and potential future of PDEs in geometric design and computer graphics.