We propose an angle-based mesh representation, which is invariant under translation, rotation, and uniform scaling, to encode the geometric details of a triangular mesh. Angle-based mesh representation consists of angle quantities defined on the mesh, from which the mesh can be reconstructed uniquely up to translation, rotation, and uniform scaling. The reconstruction process requires solving three sparse linear systems: the first system encodes the length of edges between vertices on the mesh, the second system encodes the relationship of local frames between two adjacent vertices on the mesh, and the third system defines the position of the vertices via the edge length and the local frames. From this angle-based mesh representation, we propose a quasi-angle-preserving mesh deformation system with the least-squares approach via handle translation, rotation, and uniform scaling. Several detail-preserving mesh editing examples are presented to demonstrate the effectiveness of the proposed method.

In this paper, we present an automatic approach to match correspondences on 3-D human bodies in various postures so that feature points can be automatically extracted. The feature points are very important to the establishment of volumetric parameterization around human bodies for the human-centered customization of soft-products (Trans. Autom. Sci. Eng., vol. 4, issue no. 1, pp. 11-21, 2007). For a given template human model with a set of predefined feature points, we first down-sample the input model into a set of sample points. Then, the corresponding points of these samples on the human model are identified by minimizing the distortion with the help of a series of transformations regardless of their differences in postures, scales or positions. The basic idea of our algorithm is to transform the template human body to the shape of the input model iteratively. To generate a bending invariant mapping, the initial correspondence/transformation is computed in a multidimensional scaling (MDS) embedding domain of 3-D human models, where the Euclidean distance between two samples on a 3-D model in the MDS domain corresponds to the geodesic distance between them in ℜ3. As the posture change (i.e., the body bending) of a human model can be considered as approximately isometric in the intrinsic 3-D shape, the initial correspondences established in the MDS domain can greatly enhance the robustness of our approach in body bending. Once the correspondences between the surface samples on the template model and the input model are determined after iterative transformations, we have essentially found the corresponding feature points on the input model. Finally, the locations of the based local matching step.

Sculpting deformation is a powerful tool to modify the shape of objects intuitively. However, the detail preserving problem has not been considered in sculpting deformation. In the deformation of a source object by pressing a primitive object against it, the source object is deformed while geometric details of the object should be maintained. In order to address this problem, we present a detail preserving sculpting deformation algorithm by using Laplacian coordinates. Based on the property of Laplacian coordinate, we propose two feature invariants to encode the Laplacian coordinate. Instead of mapping the source mesh to the primitive mesh, we map the smooth version of source mesh to the primitive mesh and use the Laplacian coordinates to encode the geometric details. When the smooth version of the source mesh is deformed, the Laplacian coordinates of the deformed mesh are computed for each vertex firstly and then the deformed mesh is reconstructed by solving a linear system that satisfies the reconstruction of the local details in least squares sense. Several examples are presented to show the effectiveness of the proposed approach.

This paper presents an algorithm to approximate a solid model by a hierarchical set of bounding ellipsoids having optimal shape and volume approximation errors. The ellipsoid-tree is constructed in a top-down splitting framework. Starting from the root of hierarchy the volume occupied by a given model is divided into k sub-volumes where each is approximated by a volume bounding ellipsoid and will be later subdivided into k ellipsoids for the next level in hierarchy. The difficulty for implementing this algorithm comes from how to evaluate the volume of an ellipsoid outside the given model effectively and efficiently (i.e., the outside-volume-error). A new method - analytical computation based - is presented in this paper to compute the outside-volume-error. One application of ellipsoid-tree approximation has also been given at the end of the paper.

In this paper, we investigate the direct manipulation problem of free form deformation with rational DMS spline volume (RDMS-FFD). For the weights based direct manipulation method, the solution of the weights can be achieved by solving a linear system; for the control points based method, the explicit solution of displacements of the control points can be obtained, and some properties are also derived from the results. For the constraint points inside the control lattices, we use the weights based methods; for the constraint points outside the control lattices, the control points based method is adopted. Several examples are presented to show the effectiveness of the proposed methods.

Traditional axial deformation is simple and intuitive for users to modify the shape of objects. However, unexpected twist of the object may be obtained. The use of a curve-pair allows the local coordinate frame to be controlled intuitively. However, some important geometric details may be lost and changed in the deformation process. In this paper, we present a detail-preserving axial deformation algorithm based on Laplacian coordinates. Instead of embedding the absolute coordinates into deformation space in traditional axial deformation, we transform the Laplacian coordinates at each vertex according to the transformation of local frames at the closest points on the axial curve. Then the deformed mesh is reconstructed by solving a linear system that describes the reconstruction of the local details in least squares sense. By associating a complex 3D object to a curve-pair, the object can be stretched, bend, twisted intuitively through manipulating the curve-pair, and can also be edited by means of view-dependent sketching. This method combines the advantages of axial deformation and Laplacian mesh editing. Experimental results are presented to show the effectiveness of the proposed method.

This paper presents a new approach for the mesh composition on models with arbitrary boundary topology. After cutting the needed parts from existing mesh models and putting them into the right pose, an implicit surface is adopted to smoothly interpolate the boundaries of models under composition. An interface is developed to control the shape of the implicit transient surface by using sketches to specify the expected silhouettes. After that, a localized Marching Cubes algorithm is investigated to tessellate the implicit transient surface so that the mesh surface of composed model is generated. Different from existing approaches in which the models under composition are required to have pairwise merging boundaries, the framework developed based on our techniques have the new function to fuse models with arbitrary boundary topology.

As ellipsoids have been employed in the collision handling of many applications in physical simulation and robotics systems, we present a novel algorithm for generating a bounding volume hierarchy (BVH) from a given model with ellipsoids as primitives. Our algorithm approximates the given model by a hierarchical set of optimized bounding ellipsoids. The ellipsoid-tree is constructed by a top-down splitting. Starting from the root of hierarchy, the volume occupied by a given model is divided into k sub-volumes where each is approximated by a volume bounding ellipsoid. Recursively, each sub-volume is then subdivided into ellipsoids for the next level in the hierarchy. The k ellipsoids at each hierarchy level for a sub-volume bounding is generated by a bottom-up algorithm - simply, the sub-volume is initially approximated by m spheres (m » k), which will be iteratively merged into k volume bounding ellipsoids and globally optimized to minimize the approximation error. Benefited from the anisotropic shape of primitives, the ellipsoid-tree constructed in our approach gives tighter volume bound and higher shape fidelity than another widely used BVH, sphere-tree.

Abstract In this article, a shape transformation technique is introduced for deforming objects based on a given deformation example. The example consists of two reference shapes representing two different states of an object. The reference shapes are assumed to morph from one state to the other. The evolution between the two reference shapes determines the shape transformation function. Any given objects can then be deformed by the same transformation. A continuous 4D Radial Basis Function is used to construct a density flow field (an extension of the optical flow in computer vision) representing the shape transformation of the example in 3‐space. Objects embedded in the density flow field are deformed by moving vertices of the objects along the density flow vectors. Additional parameters are introduced to control the process of the deformation. This provides explicit control on the shape of the object obtained in the deformation process. Copyright © 2007 John Wiley & Sons, Ltd.

This paper addresses the problem of volume parameterization that serves as the geometric kernel for design automation of customized free-form products. The purpose of volume parameterization is to establish a mapping between the spaces that are near to two reference free-form models, so that the shape of a product presented in free-form surfaces can be transferred from the space around one reference model to another reference model. The mapping is expected to keep the spatial relationship between the product model and reference models as much as possible. We separate the mapping into rigid body transformation and elastic warping. The rigid body transformation is determined by anchor points defined on the reference models using a least-squares fitting approach. The elastic warping function is more difficult to obtain, especially when the meshes of the reference objects are inconsistent. A three-stage approach is conducted. First, a coarse-level warping function is computed based on the anchor points. In the second phase, the topology consistency is maintained through a surface fitting process. Finally, the mapping of volume parameterization is established on the surface fitting result. Compared to previous methods, the approach presented here is more efficient. Also, benefitting from the separation of rigid body transformation and elastic warping, the transient shape of a transferred product does not give unexpected distortion. At the end of this paper, various industry applications of our approach in design automation are demonstrated. Note to Practitioners-The motivation of this research is to develop a geometric solution for the design automation of customized free-form objects, which can greatly improve the efficiency of design processes in various industries involving customized products (e.g., garment design, toy design, jewel design, shoe design, and glasses design, etc.). The products in the above industries are usually composed of a very complex geometry shape (represented by free-form surfaces), and is not driven by a parameter table but a reference object with free-form shapes (e.g., mannequin, toy, wrist, foot, and head models). After carefully designing a product around one particular reference model, it is desirable to have an automated tool for "grading" this product to other shape-changed reference objects while retaining the original spatial relationship between the product and reference models. This is called the design automation of a customized free-form object. Current commercial 3-D/2-D computer-aided design (CAD) systems, developed for the design automation of models with regular shape, cannot support the design automation in this manner. The approach in this paper develops efficient techniques for constraining and reconstructing a product represented by free-form surfaces around reference objects with different shapes, so that this design automation problem can be fundamentally solved. Although the approach has not been integrated into commercial CAD systems, the results based on our preliminary implementation are encouraging-the spatial relationship between reference models and the customized products is well preserved

An alternative approach for the simulation of metal forming process is proposed. The main idea is to simulate the shape change of the material throughout the whole process by metamorphosis, and then analyze the physical property changes among the intermediate shapes. The key advantage of the proposed approach, when compared with FEM, is that less computing power is required. Since in the metal forming process, material is expected to deform permanently, i.e. undergo plastic deformation, the volume of material is invariant over the forming process. Hence, a "three dimensional, volume conserved and parameter controlled" metamorphosis algorithm is developed. The idea is to transform first the object from the spatial domain into another domain, which provides a better control over the volume information. Interpolation is then applied in the new domain and the intermediate shapes are obtained from inverse transformation.