ABSTRACT Although numerous studies have explored relaxation and sleep aid through Autonomous Sensory Meridian Response (ASMR) videos or conventional Virtual Reality (VR) relaxation methods, the integration of VR 3D animation with ASMR and its comparison to traditional VR relaxation methods remains underexplored. To address this gap, we investigate a standardized process for creating a VR‐based ASMR animation game and its impact on triggering the ASMR tingling sensation in VR environments. We also developed a VR 3D environment game featuring four natural environments, along with one ASMR video as a control group. A comprehensive experiment was conducted to compare the effectiveness of these three relaxation methods. Forty‐seven participants aged 18–35 from Bournemouth University were recruited and divided into three experimental groups. Participants' emotional and physiological responses were monitored using both subjective questionnaires and physiological data collection that is, heart rate (HR) and electrodermal activity (EDA). Our findings show that VR‐based ASMR animation game effectively triggers the ASMR tingling experience and offers superior relaxation, sleep assistance, and emotional regulation compared to watching ASMR videos and conventional VR relaxation methods, resulting in a significant reduction in anxiety and stress, as well as increased feelings of calmness and sleepiness.

We present a Lagrangian dynamic importance Monte Carlo method without non-trivial random walks for solving the Velocity-Vorticity Poisson Equation (VVPE) in Smoothed Particle Hydrodynamics (SPH) for vortical flows. Key to our approach is the use of the Kinematic Vorticity Number (KVN) to detect vortex cores and to compute the KVN-based importance of each particle when solving the VVPE. We use Adaptive Kernel Density Estimation (AKDE) to extract a probability density distribution from the KVN for the the Monte Carlo calculations. Even though the distribution of the KVN can be non-trivial, AKDE yields a smooth and normalized result which we dynamically update at each time step. As we sample actual particles directly, the Lagrangian attributes of particle samples ensure that the continuously evolved KVN-based importance, modeled by the probability density distribution extracted from the KVN by AKDE, can be closely followed. Our approach enables effective vortical flow simulations with significantly reduced computational overhead and comparable quality to the classic Biot-Savart law that in contrast requires expensive global particle querying.

Abstract For vortex particle methods relying on SPH‐based simulations, the direct approach of iterating all fluid particles to capture velocity from vorticity can lead to a significant computational overhead during the Biot‐Savart summation process. To address this challenge, we present a Monte Carlo vortical smoothed particle hydrodynamics (MCVSPH) method for efficiently simulating turbulent flows within an SPH framework. Our approach harnesses a Monte Carlo estimator and operates exclusively within a pre‐sampled particle subset, thus eliminating the need for costly global iterations over all fluid particles. Our algorithm is decoupled from various projection loops which enforce incompressibility, independently handles the recovery of turbulent details, and seamlessly integrates with state‐of‐the‐art SPH‐based incompressibility solvers. Our approach rectifies the velocity of all fluid particles based on vorticity loss to respect the evolution of vorticity, effectively enforcing vortex motions. We demonstrate, by several experiments, that our MCVSPH method effectively preserves vorticity and creates visually prominent vortical motions.

Abstract 3D reconstruction is a long‐standing problem. Recently, a number of studies have emerged that utilize transformers for 3D reconstruction, and these approaches have demonstrated strong performance. However, transformer‐based 3D reconstruction methods tend to establish the transformation relationship between the 2D image and the 3D voxel space directly using transformers or rely solely on the powerful feature extraction capabilities of transformers. They ignore the crucial role played by deep multi‐scale representation of the object in the voxel feature domain, which can provide extensive global shape and local detail information about the object in a multi‐scale manner. In this article, we propose a novel framework TMSDNet (transformer with multi‐scale dense network) for single‐view and multi‐view 3D reconstruction with transformer to solve this problem. Based on our well‐designed combined‐transformer Block, which is canonical encoder–decoder architecture, voxel features with spatial order can be extracted from the input image, which are used to further extract multi‐scale global features in parallel using a multi‐scale residual attention module. Furthermore, a residual dense attention block is introduced for deep local features extraction and adaptive fusion. Finally, the reconstructed objects are produced with the voxel reconstruction block. Experiment results on the benchmarks such as ShapeNet and Pix3D datasets demonstrate that TMSDNet outperforms the existing state‐of‐the‐art reconstruction methods substantially.

Recent advances in NeRF-based 3D-aware GANs have achieved outstanding performance, especially in the realm of human facial representations, making projection of facial images back into their latent space superior and preferable compared to 2D GAN inversion. However, the direct application of 2DGAN inversion techniques to 3DGAN raises challenges due to potential appearance distortions and geometric inconsistences. To tackle these issues, this work presents a novel integrated framework that combines a composite inversion pipeline in both the SS and W+ spaces and integrates a contrastive-based training strategy, ensuring proficient disentanglement within the module. Moreover, we design a facial semantic manipulation technique based on dimensional analysis of the latent code, which is fully compatible with the proposed 3DGAN inversion pipeline. Comprehensive experimental validations substantiate the effectiveness of the proposed approach in executing 3d-aware face inversion and semantic editing tasks, presenting a robust technological solution for a diverse array of digital human modeling applications in the downstream.

Deep 3D morphable models (deep 3DMMs) play an essential role in computer vision. They are used in facial synthesis, compression, reconstruction and animation, avatar creation, virtual try-on, facial recognition systems and medical imaging. These applications require high spatial and perceptual quality of synthesised meshes. Despite their significance, these models have not been compared with different mesh representations and evaluated jointly with point-wise distance and perceptual metrics. We compare the influence of different mesh representation features to various deep 3DMMs on spatial and perceptual fidelity of the reconstructed meshes. This paper proves the hypothesis that building deep 3DMMs from meshes represented with global representations leads to lower spatial reconstruction error measured with L 1 and L 2 norm metrics and underperforms on perceptual metrics. In contrast, using differential mesh representations which describe differential surface properties yields lower perceptual FMPD and DAME and higher spatial fidelity error. The influence of mesh feature normalisation and standardisation is also compared and analysed from perceptual and spatial fidelity perspectives. The results presented in this paper provide guidance in selecting mesh representations to build deep 3DMMs accordingly to spatial and perceptual quality objectives and propose combinations of mesh representations and deep 3DMMs which improve either perceptual or spatial fidelity of existing methods.

Parametric curve fitting which includes curve approximation and interpolation is a critical topic in many fields, such as medical images, 3D reconstruction and data visualization, etc. To approximate or interpolate point sets with a curve, parameterization of points is usually a key step, which has a great impact on the final result. Many methods that have been proposed to tackle this problem. To our best knowledge, there is not any review paper about point set parameterization for curve fitting. In this paper, we will review the proposed techniques of parameterizing point sets for curve fitting. These methods will be divided into two categories. In the first category, the parameterization of point sets and curve fitting are treated as two separable and successive steps, specifically, the parameters obtained using some techniques in the first step are used for the curve fitting process in the second step. The two steps are solved together in the second category using more complex techniques. Methods in each of the two types will be reviewed in this paper.

Abstract The emergence of position‐based simulation approaches has quickly developed a group of new topics in the computer graphics community. These approaches are popular due to their advantages, including computational efficiency, controllability, stability and robustness for different scenarios, whilst they also have some weaknesses. In this survey, we will introduce the concept of the baseline position based dynamics (PBD) method and review the improvements and applications of PBD since 2018, including extensions for different materials and integrations with other techniques.

Abstract Binary image boundary vectorization is the process of converting raster images into vector images represented with a sequence of Bézier curves. Two main factors in reconstructing parametric curves are to approximate the underlying structure of the boundaries as much as possible while using as few curves as possible. Existing methods do not perform well when considering both of these two main factors. In this article, we mimic the process of human vectorizing image boundaries by first segmenting the boundary points into multiple segments with the corner points. For the boundary points in each segment, we adopt the bisection method to find the largest number of points, which a single curve can fit. More curves will be added if the fitting error is larger than a predefined threshold. The process is repeated until all the points in the segment are fitted, thus minimizing the number of Bézier curves. Besides, symmetric image boundaries can be detected and used to further decrease the number of curves required. Our method can also choose the optimal parameterization method case by case to further reduce the fitting error. We make a comparison with both new and classical methods and show that our method outperforms them.

Abstract Surface reconstruction is an important part in automotive styling design. Existing reconstruction methods mainly rely on the proficiency of digital modelers who manually modify the surface shape to approximate the scanned data. Apart from manual modifications, the reconstructed surfaces cannot well reflect the design intent of designers since the feature curves of clay models have not been preserved accurately. In this paper, we propose a partial differential equation (PDE) based surface reconstruction method to analytically generate optimal surfaces with Cn continuity under the constraint of the feature curves. The proposed method accurately preserves automotive feature curves and achieves automatic reconstruction of Class-A surfaces without time-consuming manual work. The effectiveness of the proposed method is demonstrated by a number of experiments that reconstruct main parts of automotive exteriors.

Online games require small data of 3D models for low storage costs, quick transmission over the Internet, and efficient geometric processing to achieve real-time performance, and new techniques of facial blendshapes to create natural facial animation. Current geometric modelling and animation techniques involve big data of geometric models and widely applied facial animation using linear interpolation cannot generate natural facial animation and create special facial animation effects. In this paper, we propose a new approach to integrate the strengths of ODE (ordinary differential equation) sweeping surfaces and Newton’s second law-based facial blendshapes to create 3D models and their animation with small data, high efficiency, and ability to create special facial effects.

Abstract We present a novel but simple physics-based method to interactively manipulate surface shapes of 3D models with $$ C^1 $$ C 1 continuity in real time. A fourth-order partial differential equation involving a sculpting force originating from elastic bending of thin plates is proposed to define physics-based deformations and achieve $$ C^1 $$ C 1 continuity at the boundary of deformation regions. In order to obtain real-time physics-based surface manipulation, we construct a mapping relationship between a deformation region in a 3D coordinate space and a unit circle on a 2D parametric plane, formulate corresponding $$ C^1 $$ C 1 continuous boundary conditions for the unit circle, and obtain a simple analytical solution to describe the physics-based deformation in the unit circle caused by a sculpting force. After that, the obtained physics-based deformation is mapped back to the 3D coordinate space, and added to the original surface to create a new surface shape with $$ C^1 $$ C 1 continuity at the boundary of the deformation region. We also develop an interactive user interface as a plug-in of the 3D modelling software package Maya to achieve real-time surface manipulation. The effectiveness, easiness, real-time performance, and better realism of our proposed method is demonstrated by testing surface deformations on several 3D models and comparing with other methods and ground-truth deformations.

Aggregate elements are ubiquitous in natural and man-made objects. Interactively authoring these elements with varying anisotropy and deformability can require high artistic skills and manual labor. To reduce input workload and enhance output quality, we present an autocomplete system that can help users distribute and align such elements over different domains. Through a brushing interface, users can place and mix a few elements, and let our system automatically populate more elements for the remaining output. Furthermore, aggregate elements often require proper direction/scalar fields for proper arrangements, but fully specifying such fields across entire domains can be difficult or inconvenient for ordinary users. To address this usability challenge, we formulate element fields that can smoothly orient all the elements based on partial user specifications without requiring full input fields in any step. We validate our prototype system with a pilot user study and show applications in design, collage, and modeling.

Abstract Virtual characters are 3D geometric models of characters. They have a lot of applications in multimedia. In this paper, we propose a new physics-based deformation method and efficient character modelling framework for creation of detailed 3D virtual character models. Our proposed physics-based deformation method uses PDE surfaces. Here PDE is the abbreviation of Partial Differential Equation, and PDE surfaces are defined as sculpting force-driven shape representations of interpolation surfaces. Interpolation surfaces are obtained by interpolating key cross-section profile curves and the sculpting force-driven shape representation uses an analytical solution to a vector-valued partial differential equation involving sculpting forces to quickly obtain deformed shapes. Our proposed character modelling framework consists of global modeling and local modeling. The global modeling is also called model building, which is a process of creating a whole character model quickly with sketch-guided and template-based modeling techniques. The local modeling produces local details efficiently to improve the realism of the created character model with four shape manipulation techniques. The sketch-guided global modeling generates a character model from three different levels of sketched profile curves called primary, secondary and key cross-section curves in three orthographic views. The template-based global modeling obtains a new character model by deforming a template model to match the three different levels of profile curves. Four shape manipulation techniques for local modeling are investigated and integrated into the new modelling framework. They include: partial differential equation-based shape manipulation, generalized elliptic curve-driven shape manipulation, sketch assisted shape manipulation, and template-based shape manipulation. These new local modeling techniques have both global and local shape control functions and are efficient in local shape manipulation. The final character models are represented with a collection of surfaces, which are modeled with two types of geometric entities: generalized elliptic curves (GECs) and partial differential equation-based surfaces. Our experiments indicate that the proposed modeling approach can build detailed and realistic character models easily and quickly.

Abstract Although different approaches of segmenting and assembling geometric models for 3D printing have been proposed, it is difficult to find any research studies, which investigate model segmentation and assembly in head‐mounted display (HMD) virtual reality (VR) environments for 3D printing. In this work, we propose a novel and interactive segmentation method for screwing assembly in the environments to tackle this problem. Our approach divides a large model into semantic parts with a screwing interface for repeated tight assembly. Specifically, after a user places the cutting interface, our algorithm computes the bounding box of the current part automatically for subsequent multicomponent semantic Boolean segmentations. Afterwards, the bolt is positioned with an improved K3M image thinning algorithm and is used for merging paired components with union and subtraction Boolean operations respectively. Moreover, we introduce a swept Boolean‐based rotation collision detection and location method to guarantee a collision‐free screwing assembly. Experiments show that our approach provides a new interactive multicomponent semantic segmentation tool that supports not only repeated installation and disassembly but also tight and aligned assembly.

Abstract Shell structures are extensively used in engineering due to their efficient load‐carrying capacity relative to material volume. However, large‐span shells require additional supporting structures to strengthen fragile regions. The problem of designing optimal stiffeners is therefore becoming a major challenge for shell applications. To address it, we propose a computational framework to design and optimize rib layout on arbitrary shell to improve the overall structural stiffness and mechanical performance. The essential of our method is to place ribs along the principal stress lines which reflect paths of material continuity and indicates trajectories of internal forces. Given a surface and user‐specified external loads, we perform a Finite Element Analysis. Using the resulting principal stress field, we generate a quad‐mesh whose edges align with this cross field. Then we extract an initial rib network from the quad‐mesh. After simplifying rib network by removing ribs with little contribution, we perform a rib flow optimization which allows ribs to swing on surface to further adjust rib distribution. Finally, we optimize rib cross‐section to maximally reduce material usage while achieving certain structural stiffness requirements. We demonstrate that our rib‐reinforced shell structures achieve good static performances. And experimental results by 3D printed objects show the effectiveness of our method.

Abstract Easy and efficient 3D modelling in virtual environments is an important and unsolved topic. This paper proposes a new modelling approach to tackle this issue. It invents convolution surface‐based brushes to directly draw 3D models in head‐mounted display‐based virtual environments. In order to maximize the efficiency, flexibility, and capacity of our proposed modelling approach, we propose three different skeleton‐based convolution surfaces to tackle different modelling tasks: point skeleton‐based convolution surfaces for metaball shapes, line skeleton‐based convolution surfaces for cylindrical shapes, and polygon skeleton‐based convolution surfaces for planar surfaces. Their combination makes 3D modelling more flexible and powerful. The high efficiency is further raised by our developed closed‐form solutions for point skeletons, ends of line skeletons, and edges of polygonal skeletons. Different user‐friendly sweeping schemes are provided to facilitate intuitive inputs for various complex shape generation. Unlike Google's Tilt Brush, which is used to create disconnected sheet‐like surfaces only, our proposed convolution surface‐based brushes can produce smoothly blended manifold surfaces, and novice users can easily learn and use them to create various interesting 3D models efficiently.

Abstract Creation of detailed character models is a very challenging task in animation production. Sketch‐based character model creation from a 3D template provides a promising solution. However, how to quickly find correct correspondences between user's drawn sketches and the 3D template model, how to efficiently deform the 3D template model to exactly match user's drawn sketches, and realize real‐time interactive modeling is still an open topic. In this paper, we propose a new approach and develop a user interface to effectively tackle this problem. Our proposed approach includes using user's drawn sketches to retrieve a most similar 3D template model from our dataset and marrying human's perception and interactions with computer's highly efficient computing to extract occluding and silhouette contours of the 3D template model and find correct correspondences quickly. We then combine skeleton‐based deformation and mesh editing to deform the 3D template model to fit user's drawn sketches and create new and detailed 3D character models. The results presented in this paper demonstrate the effectiveness and advantages of our proposed approach and usefulness of our developed user interface. Copyright © 2015 John Wiley & Sons, Ltd.

Abstract Puppetry has been a popular art form for many centuries in different cultures, which becomes a valuable and fascinating heritage assert. Traditional Chinese marionette art with over 2000 years history is one of the most representative forms offering a mixture of stage performance of singing, dancing, music, poetry, opera, story narrative and action. Apart from a set of string rules, which controls the dynamics, head carving skill is another important pillar in this art form. This paper addresses the heritage preservation of the marionette head carving by digitalizing the head models with a novel modelling technique using ordinary differential equations (ODEs). The technique has been specially tailored to suit the modelling complexity and the need of accurate description of shapes. It offers smoothly sewing ODE swept patches to represent the distinct features of a marionette head with sharp variance of local geometry. Such features otherwise are difficult to model and capture accurately, which may require a great effort and tedious handcrafting of an experienced modeller, when using other representation forms like polygons. Copyright © 2015 John Wiley & Sons, Ltd.

Abstract We present a novel approach to efficiently modelling branch structures with high‐quality meshes. Our approach has the following advantages. First, the limit surface can fit the target skeleton models as tightly as possible by reversely calculating the control vertices of subdivision surfaces. Second, high performance is achieved through our proposed analytical solutions and the parallel subdivision scheme on a graphics processing unit. Third, a smooth manifold quad‐only mesh is produced from the adopted Catmull–Clark scheme. A number of examples are given to demonstrate applications of our approach in various branch structures, such as tree branches, animal torsos, and vasculatures. Copyright © 2013 John Wiley & Sons, Ltd.

Partial differential equation-based surface modelling is a new approach of creating and manipulating three-dimensional geometric models. It uses the solution to a vector-valued partial differential equation subjected to suitably defined boundary constraints to carry out surface modeling. This paper provides a survey on this approach which summarizes various mathematical models of partial differential equation-based surface modelling, accurate and approximate analytical solutions as well as numerical solutions of the mathematical models, and the applications of partial differential equation-based surface modelling. It also discusses some future research directions of partial differential equation-based surface modelling.

Abstract This paper develops an efficient, physically based shape manipulation technique. It defines a 3D model with profile curves, and uses spine curves generated from the profile curves to control the motion and global shape of 3D models. Profile and spine curves are changed into profile and spine wires by specifying proper material and geometric properties together with external forces. The underlying physics is introduced to deform profile and spine wires through the closed form solution to ordinary differential equations for axial and bending deformations. With the proposed approach, global shape changes are achieved through manipulating spine wires, and local surface details are created by deforming profile wires. A number of examples are presented to demonstrate the applications of our proposed approach in shape manipulation. Copyright © 2010 John Wiley & Sons, Ltd.

We present a fast physics and example based skin deformation method for character animation in this paper. The main idea is to represent the skin surface and its deformations with a group of curves whose computation incurs much less computing overheads than the direct surface-based approach. A fourth order ordinary differential equation is introduced to describe the deformations of the curves between two example skin shapes which relates geometric and physical properties and externally applied forces to shape changes of the curves. The skin deformation is determined from these deformed curves. Several examples are given in this paper to demonstrate the application of the method.

In this paper, we present a method to create and manipulate free form surfaces. A surface is generated from boundary constraints through a vector-valued fourth order partial differential equation involving an externally applied force. In addition to this function, the boundary constraints and the externally applied force also serve as shape manipulation tools. The proposed approach is applied in surface creation, surface manipulation, surface rebuilding and surface animation. It can tackle the problems of tangential continuity in patch modeling and is cheap to compute. The examples given in this paper indicate that our proposed method can create and manipulate free from surfaces easily and quickly.

Abstract By simulating motion and deformation of a set of 3D characteristic curves defining surfaces, we introduce a new skin deformation technique to animate skin deformation of character models. This technique consists of two parts, representation of characteristic curves and dynamic skin deformation. The first part is to extract characteristic curves of an existing model, that is a polygon model or a NURBS model, represent the characteristic curves with time‐dependent 3D trigonometric curves, and relate the surface points of the model to these trigonometric curves. The advantage of such a treatment is that it can transform an original 3D problem into a 2D problem, making it much quicker and easier to process and compute. In the second part, a vector‐valued dynamic fourth‐order differential equation is employed to govern the behaviour of the time‐dependent 3D trigonometric curves. These dynamic differential equations incorporate both the time component and the physical properties of the material, which are given by the user. Thus, the character model can be made to animate (deform) following the user‐specified behaviour parameters. Our experiments demonstrate that this technique is able to produce realistic skin deformations efficiently and avoid the undesirable skinning problems suffered by some existing skinning techniques. Copyright © 2008 John Wiley & Sons, Ltd.

Abstract Due to their simplicity and intuitiveness, swept surfaces are widely used in many surface modelling applications. In this paper, we present a versatile swept surface technique called the boundary constrained swept surfaces. The most distinct feature is its ability to satisfy boundary constraints, including the shape and tangent conditions at the boundaries of a swept surface. This permits significantly varying surfaces to be both modelled and smoothly assembled, leading to the construction of complex objects. The representation, similar to an ordinary swept surface, is analytical in nature and thus it is light in storage cost and numerically very stable to compute. We also introduce a number of useful shape manipulation tools, such as sculpting forces, to deform a surface both locally and globally. In addition to being a complementary method to the mainstream surface modelling and deformation techniques, we have found it very effective in automatically rebuilding existing complex models. Model reconstruction is arguably one of the most laborious and expensive tasks in modelling complex animated characters. We demonstrate how our technique can be used to automate this process.

In this paper, a simple, accurate and efficient method is presented for free-form surface blending with curvature continuity. The method is based on an approximate analytical solution to a sixth order partial differential equation (PDE) subjected to blending boundary constraints. The most accurate and fast solution to a PDE is the closed form solution, which however is either very difficult to come by or does not exist. Comparison studies show that our proposed method has similar accuracy and computational efficiency to those of the closed form solution. We also examine the effects of the vector-valued parameters on the shape of blending surfaces and apply our proposed method in various surface blending examples. The outcomes indicate that the proposed approach is able to solve not only simple surface blending problems, but also complex ones which otherwise could only be solved using expensive numerical methods. In addition, this method can be extended to satisfy up to curvature continuities along 3 or 4 boundaries of a 3- or 4-sided patch and serve as an effective tool for the generation and manipulation of complex surfaces.

It has been reported a fourth order partial differential equation is effective in solving surface blending problems. In this paper, we present a new approximate solution to the fourth order partial differential equation and apply it to a number of surface blending tasks. The approximate solution consists of two parts: a blended part of boundary functions is used to accurately satisfy the original boundary conditions that define the blending surfaces; and a bivariate polynomial with zeroed boundary conditions, which is used to minimize the error of the fourth order partial differential equation. Using the developed method, a number of examples are investigated to demonstrate the applications of the proposed method in surface blending.

Blending is an interesting subject in computer graphics and CAD. Although surface blending has been well studied, to our knowledge, solid blending remains a blank research area. In this paper, we develop a partial differential equation (PDE) based solid blending method where a vector-valued fourth order PDE together with some surface boundary conditions are involved. In order to solve the partial differential equation, we propose an approximate analytical method which satisfies all the boundary conditions exactly. Unlike surface blending which only gives the information on the blending surface, solid blending represents properly not only the external surface, but also the interior of the solid. Three examples are given in the paper to demonstrate the applications of the proposed method in solid blending.

Abstract 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

Dynamic surface modeling is an important subject of geometric modeling due to their extensive applications in engineering design, entertainment and medical visualization. Many deformable objects in the real world are dynamic objects as their shapes change over time. Traditional geometric modeling methods are mainly concerned with static problems, therefore unsuitable for the representation of dynamic objects. Apart from the definition of a dynamic modeling problem, another key issue is how to solve the problem. Because of the complexity of the representations, currently the finite element method or finite difference method is usually used. Their major shortcoming is the excessive computational cost, hence not ideal for applications requiring real-time performance. We propose a representation of dynamic surface modeling with a set of fourth order dynamic partial differential equations (PDEs). To solve these dynamic PDEs accurately and efficiently, we also develop an effective resolution method. This method is further extended to achieve local deformation and produce n-sided patches. It is demonstrated that this new method is almost as fast and accurate as the analytical closed form resolution method and much more efficient and accurate than the numerical methods.

Surface representation using the solution to a partial differential equation (PDE) is an important topic of computer graphics and computer-aided design. In existing references, various free-from surfaces were created with a 4th-order PDE, which is only able to meet the tangential conditions at the surface boundaries. The need for a 6th-order PDE in surface modelling arises in two situations: one is to generate surfaces with curvature continuity and the other is to use curvature values as a user handle for surface shape manipulation. In this paper, we introduce such a 6th-order PDE for free-form surface generation and develop a finite difference method to solve this PDE. We also investigate the effects of boundary curvature and the vector-valued shape parameters on the surface shape. It was found that their variation has a strong influence on the shape of the surfaces and that, therefore, they can be used as flexible user handles.

Partial differential equations (PDEs) are powerful tools for the generation of free-form surfaces. In this paper, techniques of surface representation using PDEs of different orders are investigated. In order to investigate the real-time performance and capacity of surface generation based on the PDE method, the forms of three types of partial differential equations are put forward, which are the second, mixed and fourth order PDEs. The closed form solutions of these PDEs are derived. The advantages and disadvantages of each of them are discussed. A number of examples are given to demonstrate the use and effectiveness of the techniques.