Abstract: This paper proposes a new concept of polycube splines and develops novel modeling techniques for using the polycube splines in solid modeling and shape computing. Polycube splines are essentially a novel variant of manifold splines which are built upon the polycube map, serving as its parametric domain. Our rationale for defining spline surfaces over polycubes is that polycubes have rectangular structures everywhere over their domains except a very small number of corner points. The boundary of polycubes can be naturally decomposed into a set of regular structures, which facilitate tensor-product surface definition, GPU-centric geometric computing, and image-based geometric processing. We develop algorithms to construct polycube maps, and show that the introduced polycube map naturally induces the affine structure with a finite number of extraordinary points. Besides its intrinsic rectangular structure, the polycube map may approximate any original scanned data-set with a very low geometric distortion, so our method for building polycube splines is both natural and necessary, as its parametric domain can mimic the geometry of modeled objects in a topologically correct and geometrically meaningful manner. We design a new data structure that facilitates the intuitive and rapid construction of polycube splines in this paper. We demonstrate the polycube splines with applications in surface reconstruction and shape computing.

Abstract: The ACM Symposium on Solid and Physical Modeling and Applications is an annual international forum for the exchange of recent research results and applications of spatial modeling and computations in design, analysis and manufacturing, as well as in emerging biomedical, geophysical and other areas. Previous symposia in this series were held in Austin, Texas, 1991; Montreal, Canada, 1993; Salt Lake City, Utah, 1995; Atlanta, Georgia, 1997; Ann Arbor, Michigan in 1999 and 2001; Saarbrucken, Germany, 2002; Seattle, Washington, 2003; Genova, Italy, 2004, Cambridge, Massachusetts, 2005; and Cardiff, Wales, United Kingdom, 2006. This is the first time ACM SPM will be held in Asia. For additional information, please visit www.solidmodeling.org, the home page of The Solid Modeling Association that oversees this symposium series. The SPM symposium series started initially with the name "ACM Symposium on Solid Modeling and Applications." To emphasize the fact that solid modeling entails not only handling their geometric shapes, but also their physical properties and behaviors, the name of the symposium was expanded to The ACM Symposium on Solid and Physical Modeling and Applications (abbreviated as SPM) in 2005. SPM'07 was run in single track plenary sessions from Monday June 4 to Wednesday June 6, and was hosted by Tsinghua University, Beijing, China. Ninety four technical papers have been submitted and were reviewed by the international program committee with 87 expert reviewers from around the world. At least three external reviewers and members of the program committee reviewed and discussed each submission. A total of 25 refereed papers have been selected for plenary presentation and publication in the proceedings as full papers. Moreover, a total of 26 refereed papers have been selected for poster presentation and publication in the proceedings as short papers. The symposium program also includes three invited presentations by Gershon Elber, Herbert Edelsbrunner and Shing-Tung Yau, all leading researchers in their fields.

Abstract: We present a route planning algorithm for cable and wire layouts in complex environments. Our algorithm precomputes a global roadmap of the environment by using a variant of the probabilistic roadmap method (PRM) and performs constrained sampling near the contact space. Given the initial and the final configurations, we compute an approximate path using the initial roadmap generated on the contact space. We refine the approximate path by performing constrained sampling and use adaptive forward dynamics to compute a penetration-free path. Our algorithm takes into account geometric constraints like non-penetration and physical constraints like multi-body dynamics and joint limits. We highlight the performance of our planner on different scenarios of varying complexity.

Abstract: General haptic interaction with solid models requires an underlying physically-based model that can generate, in real-time, the forces and deformations to be rendered as a result of user interaction. In order to allow for a rich set of interactions, the physical model must support real-time topological modifications including the embedding of new elements in the model, and the introduction of cuts in the geometry. In this paper, we describe and demonstrate a physically-based framework for real-time interaction with 3D solid models discretized by finite elements. We present a model formulation that allows for fast progressive updates to be used in modeling the addition of new elements as well as dynamic inter- and intra-element changes in model connectivity. Our motivating applications have been in the area of open suturing simulations where cutting through skin and tissue, undermining skin to separate it from the underlying soft tissue, addition of sutures to close wounds, and manipulation using multiple surgical instruments simultaneously, are all tasks that must be supported. We show a new surgical simulator we recently developed to demonstrate the framework.

Abstract: We present a new method for noise removal on static and time-varying range data. Our approach predicts the restored position of a perturbed vertex using similar vertices in its neighborhood. It defines the required similarity measure in a new non-local fashion which compares regions of the surface instead of point pairs. This allows our algorithm to obtain a more accurate denoising result than previous state-of-the-art approaches and, at the same time, to better preserve fine features of the surface. Furthermore, our approach is easy to implement, effective, and flexibly applicable to different types of scanned data. We demonstrate this on several static and interesting new time-varying datasets obtained using laser and structured light scanners.

Abstract: Existing Poisson mesh editing techniques mainly focus on designing schemes to propagate deformation from a given boundary condition to a region of interest. Although solving the Poisson system in the least-squares sense distributes the distortion errors over the entire region of interest, large deformation in the boundary condition might still lead to severely distorted results. We propose to optimize the boundary condition (the merging boundary) for Poisson mesh merging. The user needs only to casually mark a source region and a target region. Our algorithm automatically searches for an optimal boundary condition within the marked regions such that the change of the found boundary during merging is minimal in terms of similarity transformation. Experimental results demonstrate that our merging tool is easy to use and produces visually better merging results than unoptimized techniques.

Abstract: The problem of distance computation arises in many applications including motion planning, CAD/CAM, dynamic simulation and virtual environments. Most prior work in this area has been restricted to separation or penetration distance computation between two objects. In this paper, we address the problem of computing a measure of distance between two configurations of a rigid or articulated model. The underlying distance metric is defined as the length of the longest displacement vector over the corresponding vertices of the model between two configurations. Our algorithm is based on Chasles theorem in Screw theory, and we show that the maximum distance can be realized only by a vertex of the convex hull of a rigid object. We use this formulation to compute the distance, and present two acceleration techniques to speed up the computation: incremental walking on the dual space of the convex hull and culling vertices on the convex hull using a bounding volume hierarchy (BVH). Our algorithm can be easily extended to articulated models by maximizing the distance over its each link and we also present culling techniques to accelerate the computation. We highlight the performance of our algorithm on many complex models and describe its application to proximity queries and motion planning.

Abstract: This paper develops a novel computational technique to define and construct powerful manifold splines with only one singular point by employing the rigorous mathematical theory of Ricci flow. The central idea and new computational paradigm of manifold splines are to systematically extend the algorithmic pipeline of spline surface construction from any planar domain to arbitrary topology. As a result, manifold splines can unify planar spline representations as their special cases. Despite their earlier success, the existing manifold spline framework is plagued by the topology-dependent, large number of singular points (i.e., |2g -- 2| for any genus-g surface), where the analysis of surface behaviors such as continuity remains extremely difficult. The unique theoretical contribution of this paper is that we devise new mathematical tools so that manifold splines can now be constructed with only one singular point, reaching their theoretic lower bound of singularity for real-world applications. Our new algorithm is founded upon the concept of discrete Ricci flow and associated techniques. First, Ricci flow is employed to compute a special metric of any manifold domain (serving as a parametric domain for manifold splines), such that the metric becomes flat everywhere except at one point. Then, the metric naturally induces an affine atlas covering the entire manifold except this singular point. Finally, manifold splines are defined over this affine atlas. The Ricci flow method is theoretically sound, and practically simple and efficient. We conduct various shape experiments and our new theoretical and algorithmic results alleviate the modeling difficulty of manifold splines, and hence, promising to promote the widespread use of manifold splines in surface and solid modeling, geometric design, and reverse engineering.

Abstract: The topic of interpolation between slices has been an intriguing problem for many years, as it offers means to visualize and investigate a three-dimensional object given only by its level sets. A slice consists of multiple non-intersecting simple contours, each defined by a cyclic list of vertices. An interpolation solution matches between a number of such slices (two or more at a time), providing means to create a closed surface connecting these slices, or the equivalent morph from one slice to another.We offer a method to incorporate the influence of more than two slices at each point in the reconstructed surface. We investigate the flow of the surface from one slice to the next by matching vertices and extracting differential geometric quantities from that matching. Interpolating these quantities with surface patches then allows a nonlinear reconstruction which produces a free-form, nonintersecting surface. No assumptions aremade about the input, such as on the number of contours in each slice, their geometric similarity, their nesting hierarchy, etc., and the proposed algorithm handles automatically all branching and hierarchical structures. The resulting surface is smooth and does not require further subdivision measures.

Abstract: 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: We define a Delaunay mesh to be a manifold triangle mesh whose edges form an intrinsic Delaunay triangulation or iDT of its vertices, where the triangulated domain is the piecewise flat mesh surface. We show that meshes constructed from a smooth surface by taking an iDT or a restricted Delaunay triangulation, do not in general yield a Delaunay mesh.We establish a precise dual relationship between the iDT and the Voronoi tessellation of the vertices of a piecewise flat (pwf) surface and exploit this duality to demonstrate criteria which ensure the existence of a proper Delaunay triangulation.

Abstract: Modeling and understanding complex non-manifold shapes is a key issue in shape analysis. Geometric shapes are commonly discretized as two- or three-dimensional simplicial complexes embedded in the 3D Euclidean space. The topological structure of a nonmanifold simplicial shape can be analyzed through its decomposition into a collection of components with a simpler topology. Here, we present a topological decomposition of a shape at two different levels, with different degrees of granularity. We discuss the topological properties of the components at each level, and we present algorithms for computing such decompositions. We investigate the relations among the components, and propose a graph-based representation for such relations.

Abstract: The concept of a β-shape has been recently proposed by extending the concept of the well-known α-shape. Since the β-shape takes full consideration of the Euclidean geometry of spherical particles, it is better suited than the (weighted) α-shape for applications using spatial queries on the system of variable sized spheres based on the Euclidean distance metric. In this paper, we present an efficient and elegant algorithm which computers a β-shape from a quasi-triangulation in O(log m + k) time in the worst case, where the quasi-triangulation has m simplicies and the boundary of β-shape consists of k simplicies. We believe that the β-shape and β-complex for a set of variable sized spheres (such as the atoms in a protein) will be very useful in the near future since the precise and efficient analysis of molecular structure can be conveniently facilitated by using these structures.

Abstract: Surface classification is one of the most fundamental problems in geometric modeling. Surfaces can be classified according to their conformal structures. In general, each topological equivalent class has infinite conformally equivalent classes.This paper introduces a novel method to classify surfaces by their conformal structures. Surfaces in the same conformal class share the same uniformization metric, which induces constant Gaussian curvature everywhere on the surface. Under the uniformization metric, each homotopy class of a closed curves on the surface has a unique geodesic. The lengths of all closed geodesics form the geodesic spectrum. The map from the fundamental group to the geodesic spectrum completely determines the conformal structure of the surface.We first compute the uniformization metric using discrete Ricci flow method, then compute the Fuchsian group generators, finally deduce the geodesic spectra from the generators in a closed form.The method is rigorous and practical. Geodesic spectra is applied as the signature of surfaces for shape comparison and classification.

Abstract: Reverse engineering, the process of obtaining a geometric CAD model from measurements obtained by scanning an existing physical model, is widely used in numerous applications, such as manufacturing, industrial design and jewellery design. In this work we propose a framework for reverse engineering objects of freeform design to obtain a fully editable feature-based CAD model that can be reproduced or modified before production. We focus on the process of detecting features on a cloud point and we present a fast method for analyzing the morphology of the surface defined by the point cloud. We compute a point wise characteristic called point concavity intensity and we use this quantity to detect regions that are then refined to object features. The proposed algorithm takes overall O(nlogn) time, where n is the cardinality of the point cloud.

Abstract: In this paper we present a method for deforming objects for graphics applications, based on the results of internal physical simulations. As driving examples, we describe in detail methods for simulating the bending of burning matches, and the crumpling of burning paper. In these cases, the small-scale changes in a chemical process result in large-scale deformations of the given object. We propose the use of a free form deformation to model such largescale deformations. Changing object properties are mapped onto the edges of a proxy object, which is then modified by treating the edges as springs. This proxy object then serves as a control structure for defining the deformation of the underlying object. The results we present are fast, controllable, and visually plausible.

Abstract: This paper considers an approach to mesh denoising based on the concept of random walks. The proposed method consists of two stages: a face normal filtering procedure, followed by a vertex position updating procedure which integrates the denoised face normals in a least-squares sense. Face normal filtering is performed by weighted averaging of normals in a neighbourhood. The weights are based on the probability of arriving at a given neighbour after a random walk of a virtual particle starting at a given face of the mesh and moving a fixed number of steps. The probability of a particle stepping from its current face to a given neighboring face is determined by the angle between the two face normals, using a Gaussian distribution whose width is adaptively adjusted to enhance the feature-preserving property of the algorithm. The vertex position updating procedure uses the conjugate gradient algorithm for speed of convergence. Analysis and experiments show that random walks of different step lengths yield similar denoising results. In particular, iterative application of a one-step random walk in a progressive manner effectively preserves detailed features while denoising the mesh very well. We observe that this approach is faster than many other feature-preserving mesh denoising algorithms.

Abstract: In this paper we show that the (co)chain complex associated with a decomposition of the computational domain, commonly called a mesh in computational science and engineering, can be represented by a block-bidiagonal matrix that we call the Hasse matrix. Moreover, we show that topology-preserving mesh refinements, produced by the action of (the simplest) Euler operators, can be reduced to multi-linear transformations of the Hasse matrix representing the complex.Our main result is a new representation of the (co)chain complex underlying field computations, a representation that provides new insights into the transformations induced by local mesh refinements. This paper is a further contribution towards bridging the subject of computer representations for solid and physical modeling---which flourished border-line between computer graphics, engineering mechanics and computer science with its own methods and data structures---under the general cover of linear algebra and algebraic topology. The main advantage of such an approach is that topology, geometry and physics may coexist in one and the same formalized framework, concurring together to define, represent and simulate the behavior of the model.Our approach is based on first principles and is general in that it applies to most representational domains that can be characterized as cell complexes, without any restrictions on their type, dimension, codimension, orientability, manifoldness, connectedness. Contrary to what might appear at first sight, the theoretical complexity of the present approach is not greater than that of current methods, provided that sparse-matrix techniques with double element access (by rows and by columns) are employed. Last but not least, our tensorbased approach is a significant step forward in achieving close integration of geometrical representations and physics-based simulations, i.e., in the concurrent modeling of shape and behavior.

Abstract: It has been observed for a long time that the operation consisting of offsetting a solid by a quantity r and then offsetting its complement by d

Abstract: We study the Voronoi diagram, under the Euclidean metric, of a set of ellipses, given in parametric representation. We use an efficient incremental algorithm and focus on the required predicates. The paper concentrates on InCircle, which is the hardest predicate: it decides the position of a query ellipse relative to the Voronoi circle of three given ellipses. We describe an exact, real-time, and complete implementation for InCircle, combining a certified numeric algorithm with algebraic computation. The numeric part leads to a real-time implementation for non-degenerate inputs. It relies on a geometric preprocessing that guarantees a unique solution in a box of parametric space, where a customized subdivision-based method approximates the Voronoi circle tracing the bisectors. Our subdivision method achieves quadratic convergence by exploiting the geometric characteristics of the problem. To achieve robustness, we develop interval-arithmetic techniques, based on the C++ package Alias. We switch to an algebraic approach for handling the degeneracies fast. Based on a different algebraic system to model InCircle, we apply real solving and resultant theory. The latter relies on certain symbolic routines which are efficiently implemented in Maple. Our approach readily generalizes to arbitrary conics. The paper concludes with experiments showing that most instances run in less than 0.1 sec, on a 2.6GHz Pentium-4, whereas degenerate cases may take up to 13 sec.

Abstract: In this paper, we describe a novel, shape-modeling approach to recovering 3D protein structures from volumetric images. The input to our method is a sequence of α-helices that make up a protein, and a low-resolution volumetric image of the protein where possible locations of α-helices have been detected. Our task is to identify the correspondence between the two sets of helices, which will shed light on how the protein folds in space. The central theme of our approach is to cast the correspondence problem as that of shape matching between the 3D volume and the 1D sequence. We model both the shapes as attributed relational graphs, and formulate a constrained inexact graph matching problem. To compute the matching, we developed an optimal algorithm based on the A*-search with several choices of heuristic functions. As demonstrated in a suite of real protein data, the shape-modeling approach is capable of correctly identifying helix correspondences in noise-abundant volumes with minimal or no user intervention.

Abstract: Motivated by the need to detect design intent in approximate boundary representation models, we give an algorithm to detect incomplete symmetries of discrete points, giving the models' potential local symmetries at various automatically detected tolerances. Here, incomplete symmetry is defined as a set of incomplete cycles which are constructed by, e.g., a set of consecutive vertices of an approximately regular polygon, induced by a single isometry. All seven 3D elementary isometries are considered for symmetry detection. Incomplete cycles are first found using a tolerance-controlled point expansion approach. Subsequently, these cycles are clustered for incomplete symmetry detection. The resulting clusters have welldefined, unambiguous approximate symmetries suitable for design intent detection, as demonstrated experimentally.

Abstract: B-spline multiplication, that is, finding the coefficients of the product B-spline of two given B-splines, is useful as an end result, in addition to being an important prerequisite component to many other symbolic computation operations on B-splines. Algorithms for B-spline multiplication standardly use indirect approaches such as nodal interpolation or computing the product of each set of polynomial pieces using various bases. The original direct approach is complicated. B-spline blossoming provides another direct approach that can be straightforwardly translated from mathematical equation to implementation; however, the algorithm does not scale well with degree or dimension of the subject tensor product B-splines. We present the Sliding Windows Algorithm (SWA), a new blossoming based algorithm for B-spline multiplication that addresses the difficulties mentioned heretofore.

Abstract: In geometry processing a refined control mesh is often used to approximate a Catmull-Clark subdivision surface (CCSS) By pushing the control points to their limit positions, a limit mesh of the subdivision surface is obtained We present a bound on the distance between a CCSS patch and its limit face in terms of the maximum norm of the second order differences of the control points The bound shows that the limit mesh may approximate the limit surface better than the corresponding control mesh in general Consequently, for a given error tolerance, fewer subdivision steps are needed if the refined control mesh is replaced with the corresponding limit mesh

Abstract: In this paper, we present a piecewise cubic, parametric surface scheme to interpolate positions and normals on a triangulated data set. For each data triangle, we fit three triangular cubic patches in a Clough-Tocher like arrangement. However, while we construct the micro-patches to meet each other C1, we only require approximate G1 continuity across macro-patches boundaries. To control the normal discontinuity on the macro-patch boundaries, neighbouring patches are constructed to interpolate the position and normals at the ends of their common boundary, as well as to have equal normals at additional points on the boundary. The resulting scheme constructs patches with similar shape to the quartic Shirman-Sequin construction, and has better shape than Peters' G1 cubic scheme on near singular data.

Abstract: Most solid models are archived using boundary representations, but they are created, edited, and optimized using high level constructive methods that rely on parameterized Boolean set operations and feature-based techniques. Downstream applications often require optimization of integral-valued performance measures over such models that include volume, mass, and energy properties, as well as more general distributed fields (stress, temperature, etc.). A key computational utility in all such applications is the computation of the sensitivity of the performance measure with respect to the parameters in the solid's construction history.We show that for a class of performance measures defined as domain integrals, the sensitivity with respect to a parameter requires integration over a subset of the solid's boundaries that is affected by that parameter. In contrast to earlier methods, the proposed approach for computing sensitivities does not require solid's boundary to remain homeomorphic, and may be used with most types of constructive representations, including CSG and feature-based representations, where the defining Boolean expression may not be known. Simplicity and effectiveness of the proposed technique are illustrated on several common shape optimization problems.

Abstract: We propose novel methods to generate a sequence of shapes that represents the pattern of morphological development or transformation of a Bezier curve. The methods utilize the intrinsic geometric structures of a Bezier curve that are derived from rib and fan decomposition (RFD) [Lee and Park 2005].Morphological development based on RFD shows a characteristic pattern of structural growth of a Bezier curve, which is the direct consequence of development path defined using fans. Morphological transformation based RFD utilizes development patterns of given curves inspired by the theory of evolutionary developmental biology: although two mature curves are quite different in shapes, we can easily find similarities in their younger shapes, which makes it easier to set up feature correspondences for blending. Further controls on base transformation and extrapolation ratio can determine the dominance of features and compensate the immaturity that may occur during the transformation.The development and transformation patterns generated with the methods have smooth and unique geometric style that cannot be generated using conventional methods based on multi-linear blending.

Abstract: Current feature models do not explicitly represent the relation between the parameters and the topology of the model. For theoretical and practical purposes, it is important to make this relation more explicit. A method is presented here that determines parameter values for which the topology of a feature model changes, i.e. the critical values of a given variant parameter. The considered feature model consists of a system of geometric constraints, relating parameters to feature geometry, and a cellular model. The cellular model partitions Euclidean space into quasi-disjoint cells, determined by the intersections of the feature geometry. Our method creates a new system of geometric constraints to relate the parameters of the model to topological entities in the cellular model. For each entity that is dependent on the variant parameter, degenerate cases are enforced by specific geometric constraints. Solving this system of constraints yields the critical parameter values. Critical values can be used to compute parameter ranges corresponding to families of objects, e.g. all parameter values which correspond to models that satisfy given topological constraints.

Abstract: This paper considers the mold design problem of computing a parting line for a complex mesh model, given a parting direction. Existing parting line algorithms are unsuitable for this case, as local variations in the orientations of the facets of such models lead to a parting line which zig-zags across the surface in an undesirable way. This paper presents a method to compute a smooth parting line which runs through a triangle band composed of triangles whose normals are approximately perpendicular to the parting direction. The skeleton of the triangle band is used to generate a structure representing distinct topological cycles, and to decompose the triangle band into singly-connected surface pieces, giving candidate paths. We choose a set of paths giving a good cycle; the final smooth parting line is then constructed by iteratively improving the quality of this cycle. Compliance in the physical material, or minor modifications to the surface itself, will ensure that such a parting line is appropriate for use.