Mechanical geometry theorem proving
Mechanical geometry theorem proving
The complexity of linear problems in fields
Journal of Symbolic Computation
Real quantifier elimination is doubly exponential
Journal of Symbolic Computation
Cyclides in computer aided geometric design
Computer Aided Geometric Design
Quantifier elimination for real algebra—the cubic case
ISSAC '94 Proceedings of the international symposium on Symbolic and algebraic computation
REDLOG: computer algebra meets computer logic
ACM SIGSAM Bulletin
Simulation and optimization by quantifier elimination
Journal of Symbolic Computation - Special issue: applications of quantifier elimination
Simplification of quantifier-free formulae over ordered fields
Journal of Symbolic Computation - Special issue: applications of quantifier elimination
Joining cyclide patches along quartic boundary curves
Proceedings of the international conference on Mathematical methods for curves and surfaces II Lillehammer, 1997
Hermite interpolation by piecewise polynomial surfaces with rational offsets
Computer Aided Geometric Design
Bezier and B-Spline Techniques
Bezier and B-Spline Techniques
A New Approach for Automatic Theorem Proving in Real Geometry
Journal of Automated Reasoning
Computational Geometry Problems in REDLOG
Selected Papers from the International Workshop on Automated Deduction in Geometry
Efficient projection orders for CAD
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
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We consider parametric families of semi-algebraic geometric objects, each implicitly defined by a first-order formula. Given an unambiguous description of such an object family and an intended alternative description we automatically construct a first-order formula which is true if and only if our alternative description uniquely describes geometric objects of the reference description. We can decide this formula by applying real quantifier elimination. In the positive case we furthermore derive the defining first-order formulas corresponding to our new description. In the negative case we can produce sample points establishing a counterexample for the uniqueness. We demonstrate our method by automatically proving uniqueness theorems for characterizations of several geometric primitives and simple complex objects. Finally, we focus on tori, characterizations of which can be applied in spline approximation theory with toric segments. Although we cannot yet practically solve the fundamental open questions in this area within reasonable time and space, we demonstrate that they can be formulated in our framework. In addition this points at an interesting and practically relevant challenge problem for automated deduction in geometry in general.