Hayong Shin

Modeling the surface swept by a generalized cutter for NC verification

Presented in this article is a procedure for representing the cutter-swept surface (CSS) of a generalized cutter in a single-valued form, z = f(x, y). The key idea is that the z-value of the CSS at a 2D point (x, y) is expressed as the sum of 1) the z-value at a point on the silhouette curve of the cutter bottom surface and 2) the incremental z-value along the cutter movement direction. Thus, the main part of the modeling method is to obtain the silhouette curve equations, which becomes a root finding problem for a quartic polynomial (when the cutter bottom surface contains a toroidal surface). The proposed method not only renders a single-valued representation for the CSS of a generalized cutter (which was not possible with the existing methods) including rounded endmill but also results in a computational scheme that is faster than the existing schemes for ball- and flat endmills.

Surface completion for shape and appearance

In this paper, we present a new surface content completion system that can effectively repair both shape and appearance from scanned, incomplete point set inputs. First, geometric holes can be robustly identified from noisy and defective data sets without the need for any normal or orientation information. The geometry and texture information of the holes can then be determined either automatically from the models’ context, or interactively from users’ selection. We use local parameterizations to align patches in order to extract their curvature-driven digital signature. After identifying the patch that most resembles each hole region, the geometry and texture information can be completed by warping the candidate region and gluing it onto the hole area. The displacement vector field for the exact alignment process is computed by solving a Poisson equation with boundary conditions. Our experiments show that the unified framework, founded upon the techniques of deformable models, local parameterization, and PDE modeling, can provide a robust and elegant solution for content completion of defective, complex point surfaces.

Elliptic Gabriel graph for finding neighbors in a point set and its application to normal vector estimation

Point-based shape representation has received increased attention in recent years, mainly due to its simplicity. One of the most fundamental operations for point set processing is to find the neighbors of each point. Mesh structures and neighborhood graphs are commonly used for this purpose. However, though meshes are very popular in the field of computer graphics, neighbor relations encoded in a mesh are often distorted. Likewise, neighborhood graphs, such as the minimum spanning tree (MST), relative neighborhood graph (RNG), and Gabriel graph (GG), are also imperfect as they usually give too few neighbors for a given point. In this paper, we introduce a generalization of Gabriel graph, named elliptic Gabriel graph (EGG), which takes an elliptic influence region instead of the circular region in GG. In order to determine the appropriate aspect ratio of the elliptic influence region of EGG, this paper also presents the analysis between the aspect ratio of the elliptic influence region and the average valence of the resulting neighborhood. Analytic and empirical test results are included.

Self-intersection Removal in Triangular Mesh Offsetting

AbstractProposed in this paper is an efficient algorithm to remove self-intersections from the raw offset triangular mesh. The resulting regular mesh can be used in shape inflation, tool path generation, and process planning to name a few. Objective is to find the valid region - set of triangles defining the outer boundary of the offset volume from the raw offset triangular mesh. Starting with a seed triangle, the algorithm grows the valid region to neighboring triangles until it reaches triangles with self-intersection. Then the region growing process crosses over the self-intersection and moves to the adjacent valid triangle. Therefore the region growing traverses valid triangles and intersecting triangles adjacent to valid triangles only. This property makes the algorithm efficient and robust, since this method omits unnecessary traversing invalid region, which usually has very complex geometric shape and contains many meaningless self-intersections.

Polygonal chain intersection

Abstract In order to find all intersections among polygonal chains, this paper presents a procedure consisting of two phases: (1) splitting polygonal chains into a minimal number of monotone chains and (2) finding intersections among the monotone chains. For the first phase, we suggest an optimal algorithm splitting polygonal chains into a minimal number of monotone chains, with an O( n+r log r) time complexity, where n is the number of line-segments in the polygonal chains and r is the number of C-subchains. For the second phase, we extend Bentley–Ottmanns sweep-line algorithm and the time complexity is O(( n+k )log m) , where k is the number of intersections, and m is the minimal number of monotone chains.

An integrated CAPP/CAM system for stamping die pattern machining

Since the early 1980s, CAPP has been expected to bridge the gap between CAD and CAM. Though numerous research works on CAPP have been reported, it is not easy to find a commercial CAPP system applicable to complicated objects with freeform shapes such as mold and die. The dependency on CAM system is holding one leg of CAPP, while the complexity of the solution space in freeform shape machining has a tight grip on the other. This paper exemplifies that these obstacles on the CAPP bridge can be overcome by the integration with CAM and by focusing on a specific application area. Major components of a staming die for the car body panel are manufactured by machining the raw stock castings, which are usually made by the lost foam casting process. Die pattern is the lost foam pattern made of Styrofoam for the raw stock casting. The industry trend is to build die patterns by CNC machining. In this paper, a highly specialized CAPP/CAM integrated system, called Generative Pattern Machining (GPM), for automatic tool paths generation to cut die pattern from the CAD model of the stamping die is described. The overall structure and the detailed steps of GPM are explained. GPM is being used by DaimlerChrysler pattern shop very successfully.

Direct Slicing of a Point Set Model for Rapid Prototyping

AbstractRecently point set model is getting increasing research attention in many geometric modeling application areas including computer graphics and CAD/CAM. This paper presents a novel approach to directly slicing point set model with the focus on making rapid prototyping part out of point set model without making any mesh or surface. Main challenge in handling point set model lies in how to interpret inter-point empty space and implicit quadric surfel is used in this research. This paper also explains how to utilize the quadric surfel for slicing the point set, so as to obtain contour curves for RP. Also described in this paper is how to extract smooth curve(s) out of the 2D point cloud obtained by slicing the 3D point set model.

Visually smooth composite surfaces for an unevenly spaced 3D data array

Abstract Fergusons composite surfaces, which are widely used in automatic surface fitting from an array of 3D points, are second-order continuous ( C 2 ), but they suffer from local flatness and bulges when the physical spacing of data points becomes uneven. This paper presents a method of constructing a G 1 composite surface which is free from any local flatness and bulges even with a very uneven spacing of data points. The proposed surface interpolation scheme consists of (1) construction of chord-length spline mesh curves from the input data, (2) conversion of each cubic curve segment to a sextic Bezier curve, (3) determination of “off-boundary” control points by using a G 1 condition, and (4) determination of “internal” control points. The surface interpolation scheme proposed in the paper has some nice features, for example: (a) it is a completely local scheme, (b) isoparametric curves of the entire surface are smooth across patch boundaries, and (c) the surface patches are all (nonrational) Bezier patches.

Beta-decomposition for the volume and area of the union of three-dimensional balls and their offsets†

Given a set of spherical balls, called atoms, in three‐dimensional space, its mass properties such as the volume and the boundary area of the union of the atoms are important for many disciplines, particularly for computational chemistry/biology and structural molecular biology. Despite many previous studies, this seemingly easy problem of computing mass properties has not been well‐solved. If the mass properties of the union of the offset of the atoms are to be computed as well, the problem gets even harder. In this article, we propose algorithms that compute the mass properties of both the union of atoms and their offsets both correctly and efficiently. The proposed algorithms employ an approach, called the Beta‐decomposition, based on the recent theory of the beta‐complex. Given the beta‐complex of an atom set, these algorithms decompose the target mass property into a set of primitives using the simplexes of the beta‐complex. Then, the molecular mass property is computed by appropriately summing up the mass property corresponding to each simplex. The time complexity of the proposed algorithm is O(m) in the worst case where m is the number of simplexes in the beta‐complex that can be efficiently computed from the Voronoi diagram of the atoms. It is known in ℝ3 that m = O(n) on average for biomolecules and m = O(n2) in the worst case for general spheres where n is the number of atoms. The theory is first introduced in ℝ2 and extended to ℝ3. The proposed algorithms were implemented into the software BetaMass and thoroughly tested using molecular structures available in the Protein Data Bank. BetaMass is freely available at the Voronoi Diagram Research Center web site.

A cocktail algorithm for planar bézier curve intersections

Abstract Presented in this paper is a new intersection algorithm between planar Bezier curves. The algorithm, named the cocktail algorithm, mixes and matches the merits of existing intersection algorithms appropriately. In the proposed approach, curves are approximated by a number of rational quadratic Bezier curve segments according to the shape characteristics of the curve. Then, the rational quadratic Bezier curve approximations are intersected using the implicitization method to produce the seeds of the numerical process. Experimental results reveal that the performance of the cocktail algorithm is superior to others for the curves with degrees higher than cubic. For cubic curves, however, the cocktail algorithm is slightly slower than the implicitization method with a hard coded resultant, but faster than others.