Inference of the positions of components in an assembly: part 2
Computer-Aided Design
System for interactive assembly modelling
Computer-Aided Design
Mechanical geometry theorem proving
Mechanical geometry theorem proving
Geometric and solid modeling: an introduction
Geometric and solid modeling: an introduction
Applications of Gro¨bner basis in non-linear computational geometry
Geometric reasoning
Automated reasoning in geometries using the characteristic set method and Gröbner basis method
ISSAC '90 Proceedings of the international symposium on Symbolic and algebraic computation
Multipolynomial resultant algorithms
Journal of Symbolic Computation
Algorithms in invariant theory
Algorithms in invariant theory
Solving Systems of Polynomial Equations
IEEE Computer Graphics and Applications
Computational geometry.
The midpoint locus of a triangle in a corner
ADG'10 Proceedings of the 8th international conference on Automated Deduction in Geometry
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The determination of a set of geometric entities that satisfy a series of geometric relations (constraints) constitutes the Geometric Constraint Satisfaction or Scene Feasibility (GCS/SF) problem. This problem appears in different forms in Assembly Planning, Constraint Driven Design, Computer Vision, etc. Its solution is related to the existence of roots to systems of polynomial equations. Previous attempts using exclusively numerical (geometry) or symbolic (topology) solutions for this problem present shortcomings regarding characterization of solution space, incapability to deal with geometric and topological inconsistencies, and very high computational expenses. In this investigation Grobner Bases are used for the characterization of the algebraic variety of the ideal generated by the set of polynomials. Properties of Grobner Bases provide a theoretical framework responding to questions about consistency, ambiguity, and dimension of the solution space. It also allows for the integration of geometric and topological reasoning. The high computational cost of Buchberger's algorithm for the Grobner Basis is compensated by the choice of a non redundant set of variables, determined by the characterization of constraints based on the subgroups of the group of Euclidean displacements SE(3). Examples have shown the advantage of using group based variables. One of those examples is discussed.