Automated Deduction in Geometry: 4th International Workshop, by Gábor Bodnár (auth.), Franz Winkler (eds.)

By Gábor Bodnár (auth.), Franz Winkler (eds.)

This booklet constitutes the completely refereed post-proceedings of the 4th foreign Workshop on computerized Deduction in Geometry, ADG 2002, held at Hagenberg fort, Austria in September 2002.

The thirteen revised complete papers offered have been rigorously chosen in the course of rounds of reviewing and development. one of the matters addressed are theoretical and methodological subject matters, similar to the answer of singularities, algebraic geometry and desktop algebra; numerous geometric theorem proving structures are explored; and functions of automatic deduction in geometry are confirmed in fields like computer-aided layout and robotics.

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Gao, S. Chou: Solving Parametric Algebraic Systems, Proc. fr Abstract. In this paper we present a classification of 3-revolute-jointed manipulators based on the cuspidal behaviour. It was shown in a previous work [16] that this ability to change posture without meeting a singularity is equivalent to the existence of a point in the workspace, such that a polynomial of degree four depending on the parameters of the manipulator and on the cartesian coordinates of the effector has a triple root. More precisely, from a partition of the parameters’space, such that in any connected component of this partition the number of triple roots is constant, we need to compute one sample point by cell, in order to have a full description, in terms of cuspidality, of the different possible configurations.

E. e. R coincides with A 6 Conclusion We give an algorithm to compute the projection over an algebraic closed field. Applying this algorithm to automatic theorem proving, we can get the weakest non-degenerate condition for which the theorem is true. In fact, we can get the sufficient and necessary condition for a geometric theorem to be false by computing the projection of a quasi variety. This algorithm also can be applied to automatic geometric formula deduction. There are more than one hundred geometric theorems which have been proved by this method.

It is enough to prove P rojxm+1 ,··· ,xn i Zero(ASi /Ji D) = i P rojxm+1 ,··· ,xn Zero(ASi /Ji D). t. t. (a1 , · · · , an ) ∈ Zero(ASi /Ji ). According to the definition of projection, (a1 , · · · , am ) ∈ P rojxm+1 ,··· ,xn Zero(ASi /Ji ), it follows that a = (a1 , · · · , am ) ∈ ∪i P rojxm+1 ,··· ,xn Zero(ASi /Ji D). It shows P rojxm+1 ,··· ,xn i Zero(ASi /Ji D) ⊂ i P rojxm+1 ,··· ,xn Zero(ASi /Ji D). The inclusion of reversal direction also can be proved by the same way. Lemma 2. t. variable ordering (x1 < x2 < · · · < xn ),AS = {A1 , · · · , As−1 }, J and J are the products of the initials of polynomials in AS and AS respectively.

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