Fast swept volume approximation of complex polyhedral models

Abstract

We present an efficient algorithm to approximate the swept volume (SV) of a complex polyhedron along a given trajectory. Given the boundary description of the polyhedron and a path specified as a parametric curve, our algorithm enumerates a superset of the boundary surfaces of SV. The superset consists of ruled and developable surface primitives, and the SV corresponds to the outer boundary of their arrangement. We approximate this boundary by using a five-stage pipeline. This includes computing a bounded-error approximation of each surface primitive, computing unsigned distance fields on a uniform grid, classifying all grid points using fast marching front propagation, iso-surface reconstruction, and topological refinement. We also present a novel and fast algorithm for computing the signed distance of surface primitives as well as a number of techniques based on surface culling, fast marching level-set methods and rasterization hardware to improve the performance of the overall algorithm. We analyze different sources of error in our approximation algorithm and highlight its performance on complex models composed of thousands of polygons. In practice, it is able to compute a bounded-error approximation in tens of seconds for models composed of thousands of polygons sweeping along a complex trajectory.

Introduction

The swept volume (SV) of a solid or a collection of surfaces in R^d is comprised of all points that it traverses during a sweeping motion. The problem of SV computation arises in different applications, including NC machining verification [10], [13], geometric modeling [16], [38], robot workspace analysis [4], [7], collision detection [31], [44], [57], maintainability studies [35], ergonomic design [3], motion planning [47], etc. A more extensive list of potential applications of SV can be found in Ref. [1].

The SV computation problem has been studied in different disciplines for more than four decades. This includes elegant work based on envelope theory, singularity theory, Lie groups, and sweep differential equations on the characterization of the problem. As a result, the mathematical formulation of SV computation is relatively well-understood.

In many applications, the main goal of SV computation is to identify and extract the boundary of SV, in particular, the outer boundary without internal voids. Most of the algorithms for computation of the boundary of SV in R^d are based, either explicitly or implicitly, on the following framework:

• Find all the boundary primitives $\text{S}$ that contribute to the outer boundary of SV.

• Compute an arrangement $\text{A(S)}$ of $\text{S}$ by performing intersection and trimming computations.

• Traverse the arrangement $\text{A(S)}$ and extract its outer boundary.¹ Here, the outer boundary of an arrangement in R^d is defined as the boundary of a d-dimensional cell, which is reachable from infinity following some continuous path, in the arrangement.

Most of the mathematical work has mainly dealt with characterizing the boundary primitives, given some assumptions on the sweeping path. There is a considerable amount of research in computational geometry on the combinatorial complexity of computing arrangements as well as on surface–surface intersection computations in geometric and solid modeling. However, the underlying combinatorial and algebraic complexity of exact SV computation is very high. Furthermore, the implementations of any algorithms for computing intersections and arrangements need to deal with accuracy and robustness issues. As a result, no practical algorithms are known for exact computation of SV for any arbitrary polyhedron sweeping along a given smooth path.

Given the underlying complexity of exact SV computation, most of the earlier work has focused on approximate techniques. Different algorithms can be characterized based on whether they are limited to 2D objects, or they only compute an image-space projection or visualization of SV from a given viewpoint, or compute a relatively coarse discretization of the boundary primitives followed by the union computation of different configurations of polyhedra along a trajectory. These algorithms are either slow for practical applications, or suffer from robustness problems, or compute a rather coarse approximation of SV.

We present an efficient algorithm to approximate the outer boundary of SVs of complex polyhedral models in R³ along a given trajectory. The algorithm initially enumerates a superset of the boundary primitives of SV, which consists of ruled and developable surfaces [54]. The ruled surface is generated by considering each edge in the original model as a ruling line and the trajectory as a directrix curve. The developable surface is obtained by applying the envelope theory to each moving triangle. Given a formulation of the boundary elements, our algorithm computes an approximation to the resulting arrangement using a five-stage pipeline. Firstly, it computes a bounded-error polygonal approximation of each surface primitive. Secondly, it samples the surface primitives by computing unsigned, directed distance fields along the vertices of a grid. Next it classifies the grid points to be either inside or outside of the surfaces to obtain the signed distance field using a novel algorithm based on marching front propagation. This is followed by iso-surface reconstruction. Finally, the algorithm performs topological refinement, taking into account some of the characterizations of the SV computation. We also present a number of acceleration techniques based on culling of surface primitives, use of interpolation-based rasterization hardware for fast computation of distance field, and a variation of fast marching level-set method for classification of grid points.

Our algorithm computes a bounded-error approximation of the SV and we analyze all sources of error. We have implemented this algorithm on a commodity-based PC with nVidia GeForce 4 graphics card, and benchmarked its performance on complex benchmarks. The underlying polyhedral models consist of thousands of triangles and are sweeping along a complex trajectory corresponding to a parametric curve. The computation of SV takes a few tens of seconds on a 2.4 GHz Pentium IV processor.

As compared to earlier approaches, the main advantages of our technique include:

• Generality: the algorithm can handle general polyhedral models, and makes no assumptions about the sweep path.

• Complex Models: the algorithm is directly applicable to complex models composed of a high number of features. Given a trajectory and a bound on the approximation error, the overall complexity increases as a linear function of the input size.

• Efficiency: the use of culling techniques and algorithms for signed distance field computation significantly improve the running time of the algorithm.

• Simplicity: the algorithm is relatively simple to implement and does not suffer from robustness problems or degeneracies.

• Good SV Approximation: our preliminary application of the algorithm to different benchmarks indicates that it can compute a good, bounded-error approximation of the boundary.

The rest of our paper is organized as follows. In Section 2, we briefly review the earlier work on SV computation. Section 3 provides the overview of our approach to SV computation. In Section 4, we present an algorithm to compute the boundary surface primitives of SV. Section 5 describes our approximation algorithm to compute the arrangement of the surface primitives using sampling and reconstruction. We analyze the performance of our algorithm in Section 6 and describe its implementation and performance in Section 7. In Section 8, we conclude our paper and present future work.