We investigate a new approach to narrowphase collision detection for rigid objects based on a Fourier series expansion. This new collision test scales with respect to accuracy (in the Hausdorff sense), which we show rigorously in the case of translational motions. Because our new form of the collision test is also a smooth inequality, it can be used as a holonomic unilateral constraint in many applications, such as path planning, rigid body dynamics, nesting or tool placement, replacing the need for more ad-hoc normal/contact-based constraint solvers. Moreover, we also show how this constraint can be directly differentiated via Fourier multipliers with only a constant factor overhead, which leads to a simple method for constructing a Jacobian for both normal forces and rotational torques.

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This paper presents a new method for evaluating boolean set operations between Binary Space Partition (BSP) trees. Our algorithm has many desirable features including both numerical robustness and O(n) output sensitive time complexity, while simultaneously admitting a straightforward implementation. To achieve these properties, we present two key algorithmic improvements. The first is a method for eliminating null regions within a BSP tree using linear programming. This replaces previous techniques based on polygon cutting and tree splitting. The second is an improved method for compressing BSP trees based on a similar approach within binary decision diagrams. The performance of the new method is analyzed both theoretically and experimentally. Given the importance of boolean set operations, our algorithms can be directly applied to many problems in graphics, CAD and computational geometry.

We describe a graphics processing unit (GPU)-based computational platform for six-dimensional configuration mapping, which is the description of the configuration space of rigid motions in terms of collision and contact constraints. The platform supports a wide range of computations in design and manufacturing, including three- and six-dimensional configuration space obstacle computations, Minkowski sums and differences, packaging problems, and sweep computations. We demonstrate dramatic performance improvements in the special case of configuration space operations that determine interference-free or containment-preserving configurations between moving solids. Our approach treats such operations as convolutions in the six-dimensional configuration space that are efficiently computed using the fast Fourier transform (FFT). The inherent parallelism of FFT algorithms facilitates a straightforward implementation of convolution on GPUs with existing and freely available libraries, making all such configuration space computations practical, and often interactive.