We report on the development of a two-dimensional geometric constraint solver. The solver is a major component of a new generation of CAD systems that we are developing based on a high-level geometry representation. The solver uses a graph-reduction directed algebraic approach, and achieves interactive speed. We describe the architecture of the solver and its basic capabilities. Then, we discuss in detail how to extend the scope of the solver, with special emphasis placed on the theoretical and human factors involved in finding a solution --- in an exponentially large search space --- so that the solution is appropriate to the application and the way of finding it is intuitive to an untrained user. 1 Introduction Solving a system of geometric constraints is a problem that has been considered by several communities, and using different approaches. For example, the symbolic computation community has considered the general problem, in the Supported in part by ONR contract N00014-90-J-...

Abstract. An approach for automatically generating loop invariants using quantifier-elimination is proposed. An invariant of a loop is hypothesized as a parameterized formula. Parameters in the invariant are discovered by generating constraints on the parameters by ensuring that the formula is indeed preserved by the execution path corresponding to every basic cycle of the loop. The parameterized formula can be successively refined by considering execution paths one by one; heuristics can be developed for determining the order in which the paths are considered. Initialization of program variables as well as the precondition and postcondition of the loop, if available, can also be used to further refine the hypothesized invariant. Constraints on parameters generated in this way are solved for possible values of parameters. If no solution is possible, this means that an invariant of the hypothesized form does not exist for the loop. Otherwise, if the parametric constraints are solvable, then under certain conditions on methods for generating these constraints, the strongest possible invariant of the hypothesized form can be generated from most general solutions of the parametric constraints. The approach is illustrated using the first-order theory of polynomial equations as well as Presburger arithmetic. 1.

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vii 1. INTRODUCTION AND RELATED WORK : : : : : : : : : : : : : : : : 1 1.1 Trends in two dimensional sketching : : : : : : : : : : : : : : : : : : : 2 1.1.1 The descriptive approach : : : : : : : : : : : : : : : : : : : : : 2 1.1.2 The constructive approach : : : : : : : : : : : : : : : : : : : : 2 1.1.3 The declarative approach : : : : : : : : : : : : : : : : : : : : : 3 1.2 Constraint solving methods : : : : : : : : : : : : : : : : : : : : : : : 5 1.2.1 Numerical constraint solvers : : : : : : : : : : : : : : : : : : : 5 1.2.2 Constructive constraint solvers : : : : : : : : : : : : : : : : : : 6 1.2.3 Propagation methods : : : : : : : : : : : : : : : : : : : : : : : 7 1.2.4 Symbolic constraint solvers : : : : : : : : : : : : : : : : : : : : 9 1.2.5 Solvers using hybrid methods : : : : : : : : : : : : : : : : : : 9 1.2.6 Other methods : : : : : : : : : : : : : : : : : : : : : : : : : : 10 1.3 The repertoire of con...

www.cs.unm.edu/~kapur Abstract. A method for generating polynomial invariants of imperative programs is presented using the abstract interpretation framework. It is shown that for programs with polynomial assignments, an invariant consisting of a conjunction of polynomial equalities can be automatically generated for each program point. The proposed approach takes into account tests in conditional statements as well as in loops, insofar as they can be abstracted to be polynomial equalities and disequalities. The semantics of each statement is given as a transformation on polynomial ideals. Merging of paths in a program is defined as the intersection of the polynomial ideals associated with each path. For a loop junction, a widening operator based on selecting polynomials up to a certain degree is proposed. The algorithm for finding invariants using this widening operator is shown to terminate in finitely many steps. The proposed approach has been implemented and successfully tried on many programs. A table providing details about the programs is given. 1

We study exact representations for offset curves and surfaces, for equal-distance curves and surfaces, and for fixed- and variable-radius blending surfaces. The representations are systems of nonlinear equations that define the curves and surfaces as natural projections from a higher-dimensional space into 3-space. We show that the systems derived by naively translating the geometric constraints defining the curves and surfaces can entail degeneracies that result in additional solutions that have no geometric significance. We characterize these extraneous solution points geometrically, and then augment the systems with auxiliary equations of a uniform structure that exclude all extraneous solutions. Thereby, we arrive at representations that capture the geometric intent of the curve and surface definitions precisely. Keywords: geometric modeling, faithful problem formulation, offsets, blends, equidistance surfaces, extraneous solutions 1. Introduction Geometric modeling uses a number o...

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Three different methods for automated geometry theorem proving -- a generalized version of Dixon resultants, Grobner 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.

Abstract. Building on a simple construction of the complex numbers and a proof of the Fundamental Theorem of Algebra, we implement, as a HOL derived inference rule, a decision method for the first order algebraic theory of C based on quantifier elimination. Although capable of solving some mildly interesting problems, we also implement a more efficient semidecision procedure for the universal fragment based on Gröbner bases. This is applied to examples including the automatic proof of some simple geometry theorems. The general and universal procedures present an interesting contrast in that the latter can exploit the finding-checking separation to achieve greater efficiency, though this feature is only partly exploited in the present implementation. 1

Dynamic geometry programs provide environments where accurate construction of geometric configurations can be done. Nevertheless, intrinsic limitations in their standard development technology mostly produce objects that are equationally unknown and so can not be further used in constructions. In this paper, we pursue the development of a geometric system that uses in the background the symbolic capabilities of two computer algebra systems, CoCoA and Mathematica. The cooperation between the geometric and symbolic modules of the software is illustrated by the computation of plane envelopes and other derived curves. These curves are described both graphically and analytically. Since the equations of these curves are known, the system allows the construction of new elements depending on them.

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