Fundamentals of interactive computer graphics
Fundamentals of interactive computer graphics
A search algorithm for motion planning with six degrees of freedom
Artificial Intelligence
Application of vector sum operator
Computer-Aided Design
The complexity of robot motion planning
The complexity of robot motion planning
Sweeping of three-dimensional objects
Computer-Aided Design
Minkowski operations for satellite antenna layout
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Polynomial/rational approximation of Minkowski sum boundary curves
Graphical Models and Image Processing
A Schur--Fréchet Algorithm for Computing the Logarithm and Exponential of a Matrix
SIAM Journal on Matrix Analysis and Applications
Representations for Rigid Solids: Theory, Methods, and Systems
ACM Computing Surveys (CSUR)
Robot Motion Planning
Approximating the Logarithm of a Matrix to Specified Accuracy
SIAM Journal on Matrix Analysis and Applications
An Algebra of Geometric Shapes
IEEE Computer Graphics and Applications
Formal engineering design synthesis
Orientation maps: techniques for visualizing rotations (a consumer's guide)
VIS '93 Proceedings of the 4th conference on Visualization '93
Implicit modeling of swept surfaces and volumes
VIS '94 Proceedings of the conference on Visualization '94
Provable surface reconstruction from noisy samples
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Image Analysis and Mathematical Morphology
Image Analysis and Mathematical Morphology
Accurate Minkowski sum approximation of polyhedral models
Graphical Models - Special issue on PG2004
Spatial Planning: A Configuration Space Approach
IEEE Transactions on Computers
Covering Minkowski sum boundary using points with applications
Computer Aided Geometric Design
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Geometric Fundamentals of Robotics
Geometric Fundamentals of Robotics
Parallel Robots
Solving inverse configuration space problems by adaptive sampling
Computer-Aided Design
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The six-dimensional space SE(3) is traditionally associated with the space of configurations of a rigid solid (a subset of Euclidean three-dimensional space R^3). But a solid itself can be also considered to be a set of configurations, and therefore a subset of SE(3). This observation removes the artificial distinction between shapes and their configurations, and allows formulation and solution of a large class of problems in mechanical design and manufacturing. In particular, the configuration product of two subsets of configuration space is the set of all configurations obtained when one of the sets is transformed by all configurations of the other. The usual definitions of various sweeps, Minkowski sum, and other motion related operations are then realized as projections of the configuration product into R^3. Similarly, the dual operation of configuration quotient subsumes the more common operations of unsweep and Minkowski difference. We identify the formal properties of these operations that are instrumental in formulating and solving both direct and inverse problems in computer aided design and manufacturing. Finally, we show that all required computations may be approximated using a fast parallel sampling method on a GPU and provide error estimates for the approximation.