Pinghai Yang

Direct Computing of Surface Curvatures for Point-Set Surfaces

Accurate computing of the curvatures of a surface from its discrete form is of fundamental importance for many graphics and engineering applications. The moving least-squares (MLS) surface from Levin [Lev2003] and its variants have been successfully used to define point-set surfaces in a variety of point cloud data based modeling and rendering applications. This paper presents a set of analytical equations for direct computing of surface curvatures from pointset surfaces based on the explicit definition from [AK04a, AK04b]. Besides the Gaussian parameter involved in the MLS definition, these analytical equations allow us to conduct direct and exact differential geometric analysis on the point-set surfaces without specifying any subjective parameters. Our experimental validation on both synthetic and real point cloud data demonstrates that such direct computing from analytical equations provides a viable approach for surface curvature evaluation for unorganized point cloud data.

A B-spline-based approach to heterogeneous objects design and analysis

The recent advancement of solid freeform fabrication, design techniques and fundamental understanding of material properties in functionally graded materials has made it possible to design and fabricate multifunctional heterogeneous objects. In this paper, we present an integrated design and analysis approach for heterogeneous object realization, which employs a unified design and analysis model based on B-spline representation and allows for direct interaction between the design and analysis model without laborious meshing operation. In the design module, a new approach for intuitively modelling of multi-material objects, termed heterogeneous lofting, is presented. In the analysis module, a novel graded B-spline finite element solution procedure is described, which gives orders of magnitude of better convergence rate in comparison with current methods, as demonstrated in several case studies. Further advantages of this approach include simplified mesh construction, exact geometry/material composition representation and easy extraction of an isomaterial surface for manufacturing process planning.

Adaptive Slicing of Moving Least Squares Surfaces: Toward Direct Manufacturing of Point Set Surfaces

Rapid advancement of 3D sensing techniques has led to dense and accurate point cloud of an object to be readily available. The growing use of such scanned point sets in product design, analysis, and manufacturing necessitates research on direct processing of point set surfaces. In this paper, we present an approach that enables the direct layered manufacturing of point set surfaces. This new approach is based on adaptive slicing of moving least squares (MLS) surfaces. Salient features of this new approach include the following: (I) It bypasses the laborious surface reconstruction and avoids model conversion induced accuracy loss. (2) The resulting layer thickness and layer contours are adaptive to local curvatures, and thus it leads to better surface quality and more efficient fabrication. (3) The curvatures are computed from a set of closed formula based on the MLS surface. The MLS surface naturally smoothes the point cloud and allows upsampling and downsampling, and thus it is robust even for noisy or sparse point sets. Experimental results on both synthetic and scanned point sets are presented.

A general, accurate procedure for calculating molecular interaction force

The determination of molecular interaction forces, e.g., van der Waals force, between macroscopic bodies is of fundamental importance for understanding sintering, adhesion and fracture processes. In this paper, we develop an accurate, general procedure for van der Waals force calculation. This approach extends a surface formulation that converts a six-dimensional (6D) volume integral into a 4D surface integral for the force calculation. It uses non-uniform rational B-spline (NURBS) surfaces to represent object surfaces. Surface integrals are then done on the parametric domain of the NURBS surfaces. It has combined advantages of NURBS surface representation and surface formulation, including (1) molecular interactions between arbitrary-shaped objects can be represented and evaluated by the NURBS model further common geometries such as spheres, cones, planes can be represented exactly and interaction forces are thus calculated accurately; (2) calculation efficiency is improved by converting the volume integral to the surface integral. This approach is implemented and validated via its comparison with analytical solutions for simple geometries. Calculation of van der Waals force between complex geometries with surface roughness is also demonstrated. A tutorial on the NURBS approach is given in Appendix A.

Direct boolean intersection between acquired and designed geometry

In this paper, a new shape modeling approach that can enable direct Boolean intersection between acquired and designed geometry without model conversion is presented. At its core is a new method that enables direct intersection and Boolean operations between designed geometry (objects bounded by NURBS and polygonal surfaces) and scanned geometry (objects represented by point cloud data). We use the moving least-squares (MLS) surface as the underlying surface representation for acquired point-sampled geometry. Based on the MLS surface definition, we derive closed formula for computing curvature of planar curves on the MLS surface. A set of intersection algorithms including line and MLS surface intersection, curvature-adaptive plane and MLS surface intersection, and polygonal mesh and MLS surface intersection are successively developed. Further, an algorithm for NURBS and MLS surface intersection is then developed. It first adaptively subdivides NURBS surfaces into polygonal mesh, and then intersects the mesh with the MLS surface. The intersection points are mapped to the NURBS surface through the Gauss-Newton method. Based on the above algorithms, a prototype system has been implemented. Through various examples from the system, we demonstrate that direct Boolean intersection between designed geometry and acquired geometry offers a useful and effective means for the shape modeling applications where point-cloud data is involved.

Topologically Enhanced Slicing of MLS Surfaces

Growing use of massive scan data in various engineering applications has necessitated research on point-set surfaces. A point-set surface is a continuous surface defined directly with a set of discrete points. This paper presents a new approach that extends our earlier work on slicing point-set surfaces into planar contours for rapid prototyping usage. This extended approach can decompose a point-set surface into slices with guaranteed topology. Such topological guarantee stems from the use of Morse theory based topological analysis of the slicing operation. The Morse function for slicing is a height function restricted to the point-set surface, an implicitly defined moving least-squares (MLS) surface. We introduce a Lagrangian multiplier formulation for critical point identification from the restricted surface. Integral lines are constructed to form Morse-Smale complex and the enhanced Reeb graph. This graph is then used to provide seed points for forming slicing contours, with the guarantee that the sliced model has the same topology as the input point-set surface. The extension of this approach to degenerate functions on point-set surface is also discussed.Copyright

NURBS based molecular force calculation

The progress of nanotechnology has made it possible to make miniature electromechanical devices of sub-micrometer scale. This means that we will be in need of design packages that can model the physical properties of objects and their interactions involved down in nanometer scale. Toward this goal, our aim in this paper is to develop a computing procedure for determining molecular interaction forces, i.e. van der Waals force, between objects of arbitrary geometry. Currently there are two types of approaches for calculating van der Waals force. The first type is analytical where analytical force equations are derived for interactions between simple geometries such as spheres and half-spaces. The second type is numerical where volume integrals or surface integrals are conducted over discretized object domains where the object boundaries are approximated by simple mesh geometries. This paper presents a numerical approach that uses non-uniform rational B-spline (NURBS) based surface integrals. The integrals are done on the parametric domains of the NURBS surfaces and Gaussian quadrature points lie exactly on the object surfaces. Salient features of this approach include: 1) Orders of magnitude in accuracy improvement is achieved over other numerical approaches; The fundamental reason for such accuracy improvement is that molecular interaction force is very sensitive to surface geometry since it falls off at the rate of inverse power of 6 ~ 7. Any geometric approximation in object discretization would lead to significant bias in the calculation result. 2) Molecular interactions between arbitrary-shaped objects can be represented and evaluated since the NURBS model can represent exactly common analytical geometries such as spheres in nano-particles and cylinders in nano-rods, and complex geometries such as corrugated sample surfaces. We demonstrate its general shape applicability by calculating van der Waals force between complex geometries such as micro-gears. Further, we give error bounds for NURBS based numerical simulation and develop an adaptive subdivision scheme to improve both calculation accuracy and efficiency.