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Although 3D Voronoi diagrams and medial axis transforms have numerous applications in biology, robotics, and manufacturing, most researchers use Voronoi diagrams of points instead of the true 3D input geometry, due to issues of robustness and scalability. In this paper, we present a robust sample-based GPU algorithm for calculating the full topology of Voronoi diagrams of non-general position spheres. Prior work demonstrated that the presence, geometry, and combinatorial basis of spheres that contribute to Voronoi vertices can be efficiently computed by shooting rays from each input sphere, mapping ray intersections with the nearest bisector surface to parametric bounding cubes, and analyzing the results. In this paper, we propose an algorithm on this parametric bounding cube to compute Voronoi edges in addition to the vertices. We successfully extract the full topology of the Voronoi diagram, including special cases such as isolated Voronoi edges that do not contain Voronoi vertices, more than three Voronoi edges emanating from a Voronoi vertex, and Voronoi edges that are shared by more than three Voronoi cells. Our GPU implementation efficiently and robustly handles all input, whether in general or non-general position, and finds all Voronoi vertices and edges, modulo the sampling density, including isolated disconnected edges.

In additive manufacturing, process parameters can have a large influence on the quality of the produced part, making it difficult to understand what tolerances are actually achievable. We present a system that can rapidly analyze part geometry and predict parallelism, perpendicularity, and angularity geometric deviations for planar surfaces, based on layer thickness and build direction. Our system can analyze multiple distinct features and their corresponding tolerances and datums to identify build directions where all specified tolerances can be achieved. This tool can be used to select an optimal build direction and to analyze whether specified tolerances are manufacturable using additive manufacturing.

We propose a new method to detect water trap regions in voids of oriented polygonal models that approximate the geometry of mechanical parts. Since water traps decrease cleaning and draining efficiency, accurately predicting such regions allows re-orienting parts to reduce manufacturing time and cost. We construct a directed graph that captures the flow of water in voids of a 3D input model, based on a fast orientation-dependent volume segmentation approach. We can quickly find the water trap regions by analyzing the directed graph. Since we take a purely geometric approach to solve this problem without employing any physical simulation, even if the geometry of the voids is complicated, we can calculate water traps quickly.

We propose a novel method to generate 2.5D CNC milling toolpaths that have characteristics designed for reducing energy usage. The energy consumed in multi-axis CNC milling is a function of the toolpath, component geometry, CNC machine construction, and the tool (cutter) itself. In this work, we focus solely on the toolpath geometry and propose a new method that, when all other factors are held constant, can result in toolpaths that require less energy for milling out the same pocket geometry. This strategy is based on selectively relaxed vector fields, specifically designed for the component to be machined. Pilot results comparing a toolpath generated using the proposed strategy to those generated using commercial CAM software indicate the potential for substantial savings in energy usage.

Computing the Minkowski sum of two arbitrary polyhedra in R^3 is difficult because of high combinatorial complexity. We present an algorithm for directly computing a voxelization of the Minkowski sum of two closed watertight input polyhedra for applications such as path planning that do not require a boundary representation as output. We introduce a new decomposition formula for computing the Minkowski sum and prove its correctness. We describe an efficient Graphics Processing Unit (GPU) implementation of the algorithm using stencil shadow volumes to create a solid voxelization of the Minkowski sum.

We present an algorithm for improving the rendering appearance of CAD models with trimmed freeform surfaces when evaluated on graphics processing units (GPUs). Rendering on client GPUs allows mechanical CAD to embrace cloud computing by storing a single auto-synchronized model file in the cloud and transferring only minimal data (control points, trim curves, etc.) to the client nodes for local evaluation/rendering. However, current parallel algorithms that directly evaluate and render trimmed surfaces by masking the trims, without tessellating along the trim curves, suffer from ''cracks'' along the trim boundaries. We have developed a hybrid CPU-GPU algorithm to remove these artifacts in the rendering stage for a smooth, color- and shading-matched appearance. After dynamically detecting the cracks, our algorithm selectively fills in the affected pixels using a GPU fragment program, while avoiding artifacts at silhouettes. We have implemented this algorithm to demonstrate improvements in the appearance of solid models directly evaluated and rendered on the GPU.

We present a GPU algorithm for computing the directed Hausdorff distance between two NURBS surfaces. The algorithm is based on sampling of one surface, and performing numerical iterations on the GPU to compute the minimal distance from each sample to the other surface. An error analysis for the Hausdorff distance computations is performed, based on bounds on the NURBS surfaces. We compare a CUDA implementation of our algorithm to existing methods, demonstrating that the new method addresses limitations of previous hierarchical culling methods such as the sensitivity to the position of the inputs.

We present a parallel GPU-accelerated algorithm for computing the directed Hausdorff distance from one NURBS surface to another, within a bound. We make use of axis-aligned bounding-box hierarchies that bound the NURBS surfaces to accelerate the computations. We dynamically construct as well as traverse the bounding-box hierarchies for the NURBS surfaces using operations that are optimized for the GPU. To compute the Hausdorff distance, we traverse this hierarchy after culling bounding-box pairs that do not contribute to the Hausdorff distance. Our contribution includes two-sided culling tests that can be performed in parallel using the GPU. The culling, based on the minimum and maximum distance ranges between the bounding boxes, eliminates bounding-box pairs from both surfaces that do not contribute to the Hausdorff distance simultaneously. We calculate accuracy bounds for our computed Hausdorff distance based on the curvature of the surfaces. Our algorithm runs in real-time with very small guaranteed error bounds for complex NURBS surfaces. Since we dynamically construct our bounding-box hierarchy, our algorithm can be used to interactively compute the Hausdorff distance for models made of dynamic deformable surfaces. Highlights? GPU algorithm to compute directed Hausdorff distance between NURBS surfaces. ? GPU bounding-box hierarchy traversal with selective culling of surface sub-patches. ? Novel culling tests to cull bounding-box pairs that do not contribute to HD. ? Tight range for the HD and the surface locations where the HD is within this bound. ? Interactive computation of the HD between dynamic deformable surface models.

We present a new approach for computing the voxelized Minkowski sum (excluding any enclosed voids) of two polyhedral objects using programmable Graphics Processing Units (GPUs). We first cull out surface primitives that will not contribute to the final boundary of the Minkowski sum, analyzing and adaptively bounding the rounding errors of the culling algorithm to solve the floating point error problem. The remaining surface primitives are then rendered to depth textures along six orthogonal directions to generate an initial solid voxelization of the Minkowski sum. Finally we employ fast flood fill to find all the outside voxels. We generate both solid and surface voxelizations of Minkowski sums without enclosed voids and support high volumetric resolution of 1024^3 with low video memory cost. The whole algorithm runs on the GPU and is at least one order of magnitude faster than existing boundary representation (B-rep) based algorithms. It avoids the large number of 3D Boolean operations needed in most existing algorithms and is easy to implement. The voxelized Minkowski sums can be used in a variety of applications including motion planning and penetration depth computation.

We present algorithms for computing accurate moments of solid models that are represented using multiple trimmed NURBS surfaces and triangles. Our algorithms make use of programmable Graphics Processing Units (GPUs) to accelerate the moment computations. For NURBS surfaces, we evaluate the surface coordinates and normals accurately, with theoretical bounds, using our GPU NURBS evaluator. We have developed a framework that makes use of this data to evaluate surface integrals of trimmed NURBS surfaces in real time. Since typical solid models also contain flat planes that are usually tessellated into triangles, our framework also includes GPU acceleration of the moment contributions of such flat faces. Using our framework, we can compute volume and moments of solid models with theoretical guarantees. Our algorithms support local geometry changes, which is useful for providing interactive feedback to the designer while the solid model is being designed. We can compute the center of mass and check for stability of the solid model interactively. Other applications of such real-time moment computation include deformation modeling, animation, and physically based simulations.

Given a triangular mesh defining the geometry of a 3D workpiece filled with water, we propose an algorithm to test whether, for an arbitrary given axis, the workpiece will be completely drained under gravity when the rotation axis is set parallel to the ground and the workpiece is rotated around the axis. Observing that all water traps contain a concave vertex, we solve our problem by constructing and analyzing a directed ''draining graph'' whose nodes correspond to concave vertices of the geometry and whose edges are set according to the transition of trapped water when we rotate the workpiece around the given axis. Our algorithm to test whether or not a given rotation axis drains the workpiece outputs a result in about a second for models with more than 100,000 triangles after a few seconds of preprocessing.

This paper presents a new unified and optimized method for evaluating and displaying trimmed NURBS surfaces using the Graphics Processing Unit (GPU). Trimmed NURBS surfaces, the de facto standard in commercial mechanical CAD modeling packages, are currently being tessellated into triangles before being sent to the graphics card for display since there is no native hardware support for NURBS. Other GPU-based NURBS evaluation and display methods either approximated the NURBS patches with lower degree patches or relied on specific hard-coded programs for evaluating NURBS surfaces of different degrees. Our method uses a unified GPU fragment program to evaluate the surface point coordinates of any arbitrary degree NURBS patch directly, from the control points and knot vectors stored as textures in graphics memory. This evaluated surface is trimmed during display using a dynamically generated trim-texture calculated via alpha blending. The display also incorporates dynamic Level of Detail (LOD) for real-time interaction at different resolutions of the NURBS surfaces. Different data representations and access patterns are compared for efficiency and the optimized evaluation method is chosen. Our GPU evaluation and rendering speeds are more than 40 times faster than evaluation using the CPU.

We present new programmable graphics hardware accelerated algorithms to test the 2-moldability of geometric parts and assist with part redesign. These algorithms efficiently identify and graphically display undercuts as well as minimum and insufficient draft angles. Their running times grow only linearly with respect to the number of facets in the solid model, making them efficient subroutines for our algorithms that test whether a tessellated CAD model can be manufactured in a two-part mold. We have developed and implemented two such algorithms to choose candidate directions to test for 2-moldability using accessibility analysis and Gauss maps. The efficiency of these algorithms lies in the fact that they identify groups of candidate directions such that if any one direction in the group is undercut-free, all are, or if any one is not undercut-free, none are. We examine trade-offs between the algorithms' speed, accuracy, and whether they guarantee that an undercut-free direction will be found for a part if one exists.