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Given a finite set of varieties in some nonsingular affine variety W. They are normal crossing if and only if at every point of W there is a regular system of parameters such that each variety can be defined locally at the point by a subset of this parameter system. In this paper we present two algorithms to test this property. The first one is developed for hypersurfaces only, and it has a straightforward...
In this paper, we present an algorithm to compute the projection of a quasi variety over an algebraic closed field. Based on the algorithm, we give a method to prove geometric theorem mechanically, and the non-degenerate conditions that we get by the method are proved to be the ”weakest”, i.e. the geometric theorem is true if and only if these non-degenerate conditions are satisfied. A method for...
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...
We introduce a software package, MMP/Geometer, developed by us to automate some of the basic geometric activities including geometric theorem proving, geometric theorem discovering, and geometric diagram generation. As a theorem prover, MMP/Geometer implements Wu’s method for Euclidean and differential geometries, the area method and the geometric deductive database method. With these methods, we...
Formalized proof schemes are the starting point for testing, comparing, and benchmarking of different geometry theorem proving approaches and provers. To automatize such tests it is desirable to collect a common data base of proof schemes, and to develop tools to extract examples, prepare them for input to different provers, and run them “in bulk”. The main drawback so far of special collections,...
The structural rigidity property, a generalisation of Laman’s theorem which characterises generically rigid bar frameworks in 2D, is generally considered a good heuristic to detect rigidities in geometric constraint satisfaction problems (GCSPs). In fact, the gap between rigidity and structural rigidity is significant and essentially resides in the fact that structural rigidity does not take geometric...
Cayley algebra and bracket algebra are important approaches to invariant computing in projective and affine geometries, but there are some difficulties in doing algebraic computation. In this paper we show how the principle “breefs” – bracket-oriented representation, elimination and expansion for factored and shortest results, can significantly simplify algebraic computations. We present several typical...
We give a theorem which is of significance with respect to the existence of fast algebraic methods for the solution of planar CAD diagrams. By an algebraic method we mean a solution scheme for a family of generically dimensioned graphs which rests on the usual operations of arithmetic together with arbitrary root extraction. The result establishes the non-existence of any such algebraic method whenever...
In this survey, we describe the following different aspects of modeling multidimensional point sets (shapes) using real-valued functions of several variables: algebraic system as a formal framework; representation of shapes, operations, and relations using real-valued functions, internal representation of the modeling system; specialized language for function-based modeling, and model extension to...
We present a new method for constructing a low degree C1 implicit spline representation of a given parametric planar curve. To ensure the low degree condition, quadratic B-splines are used to approximate the given curve via orthogonal projection in Sobolev spaces. Adaptive knot removal, which is based on spline wavelets, is used to reduce the number of segments. The spline segments are...
In this paper we present an algorithm to prove geometric theorems with the Gröbner basis method in Grassman algebra based on a coordinate-free approach.
GEOTHER provides an environment for handling and proving theorems in geometry automatically. In this environment, geometric theorems are represented by means of predicate specifications. Several functions are implemented that allow one to translate the specification of a geometric theorem into an English or Chinese statement, into algebraic expressions, and into a logic formula automatically. Geometric...
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