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Three different methods for automated geometry theorem proving—a generalized version of Dixon resultants, Gröbner bases and characteristic sets—are reviewed. The main focus is, however, on the use of the generalized Dixon resultant formulation for solving geometric problems and determining geometric quantities.
Dixon's resultant method is an efficient way of simultaneously eliminating several variables from a system of nonlinear polynomial equations at a time. However, the method only works for systems of n + 1 generic n-degree polynomials in n variables and does not work for most algebraic and geometric problems. In this paper, by using techniques from pseudoinverse theory and linear transformations, the...
We solve algorithmic geometrical problems in real 3-space or the real plane arising from applications in the area of cad, computer vision, and motion planning. The problems include parallel and central projection problems, shade and cast shadow problems, reconstruction of objects from images, offsets of objects, Voronoi diagrams of a finite family of objects, and collision of moving objects. Our tools...
In this paper a probabilistic approach to automated theorem proving in elementary geometry is shown. Bounds on the effective Hilbert Nullstellensatz and on the degree of a Ritt characteristic set are used together with Schwartz's probabilistic results on polynomial identities.
Computational synthetic geometry is an approach to solving geometric problems on a computer, in which the quantities appearing in the equations are all covariant under the corresponding group of transformations, and hence possess intrinsic geometric meanings. The natural covariant algebra of metric vector spaces is called Clifford algebra, and it includes Gibbs' vector algebra as a special...
In this paper we report on our recent study of Clifford algebra for geometric reasoning and its application to problems in computer vision. A general framework is presented for construction and representation of geometric objects with selected rewrite rules for simplification. It provides a mechanism suitable for devising methods and software tools for geometric reasoning and computation. The feasibility...
In this survey paper we give the basic properties of Grassmann algebras, present a generalised theory of area from a Grassmann algebra perspective, present a version for Grassmann algebras of the Buchberger algorithm, and give examples of computation and deduction in Grassmann geometry.
We present a complete method which can be used to produce short and human readable proofs for a class of constructive geometry statements in non-Euclidean geometries. The method is a substantial extension of the area method for Euclidean geometry. The method is an elimination algorithm which is similar to the variable elimination method of Wu used for proving geometry theorems. The difference is that...
A plane algebraic curve is given as the zeros of a bivariate polynomial. However, this implicit representation is badly suited for many applications, for instance in computer aided geometric design. What we want in many of these applications is a rational parametrization of an algebraic curve. There are several approaches to deciding whether an algebraic curve is rationally parametrizable and if so...
The flat central configurations of four planet motions are investigated with Wu's elimination method. We obtain 12 collinear central configurations and a necessary condition for determining flat but noncollinear central configurations. We also prove that the number of central configurations in planet motions of 4 bodies is finite under the condition that the masses and angular velocity of the planets...
This paper presents a new framework for merging reasoning and algebraic calculus in elementary geometry. This approach is based on the use of algebraic constraints in clause-based calculi. These constraints are considered as contexts for reasoning. It allows to introduce new inference rules and to use the powerful algebraic methods which have been successful in geometry theorem proving. The semantics...
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