An algorithm for automatic checking of exercises in a dynamic geometry system: iGeom

One of the key issues in e-learning environments is the possibility of creating and evaluating exercises. However, the lack of tools supporting the authoring and automatic checking of exercises for specifics topics (e.g., geometry) drastically reduces advantages in the use of e-learning environments on a larger scale, as usually happens in Brazil. This paper describes an algorithm, and a tool based on it, designed for the authoring and automatic checking of geometry exercises. The algorithm dynamically compares the distances between the geometric objects of the student’s solution and the template’s solution, provided by the author of the exercise. Each solution is a geometric construction which is considered a function receiving geometric objects (input) and returning other geometric objects (output). Thus, for a given problem, if we know one function (construction) that solves the problem, we can compare it to any other function to check whether they are equivalent or not. Two functions are equivalent if, and only if, they have the same output when the same input is applied. If the student’s solution is equivalent to the template’s solution, then we consider the student’s solution as a correct solution. Our software utility provides both authoring and checking tools to work directly on the Internet, together with learning management systems. These tools are implemented using the dynamic geometry software, iGeom, which has been used in a geometry course since 2004 and has a successful track record in the classroom. Empowered with these new features, iGeom simplifies teachers’ tasks, solves non-trivial problems in student solutions and helps to increase student motivation by providing feedback in real time.     

Symbolic data analysis tools for recommendation systems

Recommender systems have become an important tool to cope with the information overload problem by acquiring data about user behavior. After tracing the user’s behavior, through actions or rates, computational recommender systems use information- filtering techniques to recommend items. In order to recommend new items, one of the three major approaches is generally adopted: content-based filtering, collaborative filtering, or hybrid filtering. This paper presents three information-filtering methods, each of them based on one of these approaches. In our methods, the user profile is built up through symbolic data structures and the user and item correlations are computed through dissimilarity functions adapted from the symbolic data analysis (SDA) domain. The use of SDA tools has improved the performance of recommender systems, particularly concerning the find good items task measured by the half-life utility metric, when there is not much information about the user.     

Computing parameter ranges in constructive geometric constraint solving: Implementation and correctness proof

In parametric design, changing values of parameters to get different solution instances to the problem at hand is a paramount operation. One of the main issues when generating the solution instance for the actual set of parameters is that the user does not know in general which is the set of parameter values for which the parametric solution is feasible. Similarly, in constraint-based dynamic geometry, knowing the set of critical points where construction feasibility changes would allow to avoid unexpected and unwanted behaviors.We consider parametric models in the Euclidean space with one internal degree of freedom. In this scenario, in general, the set of values of the variant parameter for which the parametric model is realizable and defines a valid shape is a set of intervals on the real line.In this work we report on our experiments implementing the van der Meiden Approach to compute the set of parameter values that bound intervals for which the parametric object is realizable. The implementation is developed on top of a constructive, ruler-and-compass geometric constraint solver. We formalize the underlying concepts and prove that our implementation is correct, that is, the approach exactly computes all the feasible interval bounds.     

Searching the Solution Space in Constructive Geometric Constraint Solving with Genetic Algorithms

Geometric problems defined by constraints have an exponential number of solution instances in the number of geometric elements involved. Generally, the user is only interested in one instance such that besides fulfilling the geometric constraints, exhibits some additional properties. Selecting a solution instance amounts to selecting a given root every time the geometric constraint solver needs to compute the zeros of a multi valuated function. The problem of selecting a given root is known as the Root Identification Problem.In this paper we present a new technique to solve the root identification problem. The technique is based on an automatic search in the space of solutions performed by a genetic algorithm. The user specifies the solution of interest by defining a set of additional constraints on the geometric elements which drive the search of the genetic algorithm. The method is extended with a sequential niche technique to compute multiple solutions. A number of case studies illustrate the performance of the method.     

Fault detection and isolation in nonlinear systems

This paper deals with fault detection and identification in dynamic systems when the system dynamics can be modeled by smooth nonlinear differential equations including affine, bilinear or linear parameter varying (LPV) systems. Two basic approaches will be considered, these apply differential algebraic and differential geometric tools.In the differential algebraic approach the state elimination methods will be used to derive nonlinear parity relations. In the specific case when a reconstruction of the fault signal is needed the dynamic inversion based approach will be investigated. This approach will also be studied from geometric point of view. The geometric approach, as proposed by Isidori and De Persis, is suitable to extend the detection filter and unknown input observer design approaches (well elaborated for LTI systems) to affine nonlinear systems.Beyond the development of the theory of fault detection and identification it is equally important to offer computable methods and to analyze the robustness properties against uncertainties. Both the observer based and the inversion based approaches will be elaborated for LPV systems that may offer computational tools inherited from linear systems and also allow to design for robustness utilizing results from robust filtering and disturbance attenuation.     

科技文档中基于约束的平面几何作图

为了提高科技文档中几何作图的效率,在科技文档字处理软件ScienceWord中实现了一种基于约束的平面几何作图系统.为了构建基于约束的作图系统,对约束的相关理论作了研究,讨论了该作图系统的总体设计框架和构建约束作图系统的一般步骤,介绍了系统中建立元素之间约束的用户界面设计.由于系统充分考虑了图形元素之间的几何关系,用户不必求助于别的软件就能在科技文档中高效地绘制出各种复杂的几何图形.     

The Reachability Problem in Constructive Geometric Constraint Solving Based Dynamic Geometry

An important issue in dynamic geometry is the reachability problem that asks whether there is a continuous path that, from a given starting geometric configuration, continuously leads to an ending configuration. In this work we report on a technique to compute a continuous evaluation path, if one exists, that solves the reachability problem for geometric constructions with one variant parameter. The technique is developed in the framework of a constructive geometric constraint-based dynamic geometry system, uses the A^??? algorithm and minimizes the variant parameter arc length.     

Formulating and solving parametric design problems involving non-interference constraints

Improvements in computer-aided design (CAD) tools can significantly increase designer productivity, since the ability to explore a variety of possible designs quickly and effectively is essential for a designer. Using an optimization tool, systematic exploration of design spaces can be achieved readily. The general goal of the work presented here is to aid design by combining the strengths of optimization techniques with those of CAD systems. The specific objective of this paper is to introduce goal directed geometry (GDG) as a computational framework for parametric design, aiding the formulation of engineering problems with geometric considerations and their solution with a multi-objective optimization package. Using GDG, What if questions can be posed and answered in a systematic fashion. Specific issues to be addressed include the development of a general parametric design problem formulation, development of static and dynamic geometric non-interference constraints for use in this formulation, and investigation of the efficacy of the adaptive linear programming (ALP) multiobjective optimization algorithm in solving such problems. Two examples are presented, one each to illustrate the use of the static and dynamic non-interference constraints. Results demonstrate that the GDG formulation can be applied readily to a wide variety of parametric design problems. Additionally, the ALP algorithm successfully navigates around geometric constraints, although care must be taken when linearizing highly non-linear design spaces.     

Design and Fabrication of Free-Form Reciprocal Structures

Due to their non-hierarchical nature, the geometry of reciprocal assemblies cannot be described conveniently with the available CAD modelling tools or by hierarchical, associative parametric modellers. The geometry of a network of reciprocally connected elements is a characteristic that emerges, bottom-up, from the complex interaction between all the elements’ shape, topology and position, and requires numerical solution of the elements’ geometric compatibility. A computational method, the “Reciprocalizer”, has been developed by the authors to predict and control the geometry of large networks of reciprocally connected elements, and it has been now perfected and included in a larger procedure that can be regarded as an extremely flexible and capable design tool for the generation of free-form reciprocal structures. The design tool has been applied for the design and realization of a free-form structure composed of 506 round, un-notched wooden elements with a diameter of 22 mm. This paper focuses on the geometry of reciprocal systems and the unique issues of fabrication posed by such assemblies.     

Spatial Formulation of Elastic Multibody Systems with Impulsive Constraints

The problem of modeling the transient dynamics ofthree-dimensional multibody mechanical systems which encounter impulsiveexcitations during their functional usage is addressed. The dynamicbehavior is represented by a nonlinear dynamic model comprising a mixedset of reference and local elastic coordinates. The finite-elementmethod is employed to represent the local deformations ofthree-dimensional beam-like elastic components by either a finite set ofnodal coordinates or a truncated set of modal coordinates. Thefinite-element formulation will permit beam elements with variablegeometry. The governing equations of motion of the three-dimensionalmultibody configurations will be derived using the Lagrangianconstrained formulation. The generalized impulse-momentum-balance methodis extended to accommodate the persistent type of the impulsiveconstraints. The developed formulation is implemented into a multibodysimulation program that assembles the equations of motion and proceedswith its solution. Numerical examples are presented to demonstrate theapplicability of the developed method and to display its potential ingaining more insight into the dynamic behavior of such systems.     

Probabilistic visibility evaluation using geometry proxies

Evaluating the visibility between two points is a fundamental problem for ray‐tracing and path‐tracing algorithms. Ideally, visibility computations are organized such that a minimum number of geometric primitives need to be checked for each ray. Replacing complex geometric shapes by a simpler set of primitives is one strategy to control the amount of intersection calculations. However, approximating the original geometry introduces inaccuracies in e.g. shadow regions when shadow rays are intersected with the approximate geometry. This paper presents a theoretical framework for probabilistic visibility evaluation. When intersecting a shadow ray with the scene, we randomly select the original geometry, the approximated geometry, or one of several correction terms, to be tested. Not all shadow rays will therefore intersect the original geometry, but our method is able to produce unbiased images that converge to the correct solution. Although probabilistic visibility evaluation is an experimental idea, we show several example scenes that highlight the potential for future improvements.     

Parameter identifiability of nonlinear systems: the role of initial conditions

Identifiability is a fundamental prerequisite for model identification; it concerns uniqueness of the model parameters determined from the input-output data, under ideal conditions of noise-free observations and error-free model structure. In the late 1980s concepts of differential algebra have been introduced in control and system theory. Recently, differential algebra tools have been applied to study the identifiability of dynamic systems described by polynomial equations. These methods all exploit the characteristic set of the differential ideal generated by the polynomials defining the system. In this paper, it will be shown that the identifiability test procedures based on differential algebra may fail for systems which are started at specific initial conditions and that this problem is strictly related to the accessibility of the system from the given initial conditions. In particular, when the system is not accessible from the given initial conditions, the ideal I having as generators the polynomials defining the dynamic system may not correctly describe the manifold of the solution. In this case a new ideal that includes all differential polynomials vanishing at the solution of the dynamic system started from the initial conditions should be calculated. An identifiability test is proposed which works, under certain technical hypothesis, also for systems with specific initial conditions.     

Managing search complexity in linguistic geometry

This paper is a new step in the development of linguistic geometry. This formal theory is intended to discover and generalize the inner properties of human expert heuristics, which have been successful in a certain class of complex control systems, and apply them to different systems. In this paper, we investigate heuristics extracted in the form of hierarchical networks of planning paths of autonomous agents. Employing linguistic geometry tools the dynamic hierarchy of networks is represented as a hierarchy of formal attribute languages. The main ideas of this methodology are shown in the paper on two pilot examples of the solution of complex optimization problems. The first example is a problem of strategic planning for the air combat, in which concurrent actions of four vehicles are simulated as serial interleaving moves. The second example is a problem of strategic planning for the space comb of eight autonomous vehicles (with interleaving moves) that requires generation of the search tree of the depth 25 with the branching factor 30. This is beyond the capabilities of modern and conceivable future computers (employing conventional approaches). In both examples the linguistic geometry tools showed deep and highly selective searches in comparison with conventional search algorithms. For the first example a sketch of the proof of optimality of the solution is considered.     

Interaction with the Boyer-Moore theorem prover: A tutorial study using the arithmetic-geometric mean theorem

There are many papers describing problems solved using the Boyer-Moore theorem prover, as well as papers describing new tools and functionalities added to it. Unfortunately, so far there has been no tutorial paper describing typical interactions that a user has with this system when trying to solve a nontrivial problem, including a discussion of issues that arise in these situations. In this paper we aim to fill this gap by illustrating how we have proved an interesting theorem with the Boyer-Moore theorem prover: a formalization of the assertion that the arithmetic mean of a sequence of natural numbers is greater than or equal to their geometric mean. We hope that this report will be of value not only for (non-expert) users of this system, who can learn some approaches (and tricks) to use when proving theorems with it, but also for implementors of automated deduction systems. Perhaps our main point is that, at least in the case of Nqthm, the user can interact with the system without knowing much about how it works inside. This perspective suggests the development of theorem provers that allow interaction that is user oriented and not system developer oriented. This research was supported in part by ONR Contract N00014-94-C-0193. The views and conclusions contained in this document are those of the author(s) and should not be interpreted as representing the official policies, either expressed or implied, of Computational Logic, Inc., the Office of Naval Research, or the U.S. government.     

Adaptive maneuvering, with experiments, for a model ship in a marine control laboratory

The maneuvering problem involves two tasks. The first, called the geometric task, is to force the system output to converge to a desired path continuously parametrized by a scalar θ. The second task, called the dynamic task, is to satisfy a desired dynamic behavior along the path. In this paper, this dynamic behavior is further specified as a speed assignment for θ(t). While the main concern is to satisfy the geometric task, the dynamic task ensures that the system output follows the path with the desired speed. An adaptive recursive design technique is developed for a parametrically uncertain nonlinear plant describing the dynamics of a ship. First the geometric part of the problem is solved. Then an update law is constructed that bridges the geometric design with the dynamic task. The design procedure is performed and tested by several experiments for a model ship in a marine control laboratory.     

Reasoning with geometric information in digital space

Concurrency requires consistency and correctness. Isothetic rectangles can be used as a geometrical technique to verify a safe and deadlock free schedule for concurrent nodes. However, the known algorithms for concurrency using isothetic rectangles require the prior knowledge of the system behavior. We provide a new mechanism to use isothetic rectangles without this limitation. The discrete nature of isothetic rectangles provides an opportunity for inter-diagrammatic reasoning. Inter-Diagrammatic Reasoning (IDR) can be easily computed on a parallel machine, and has a complexity of O(n) for most of the iso-rectangles problems where the best known algorithm was O(n log n) in Euclidean geometry. This new framework will also allow dynamic mode of operation in calculating the closure of a set of iso-rectangles; rather than restricting the solution to static systems where all required resources must be reserved in advance.     

Bifurcation, pre- and post-buckling analysis of frame structures

A procedure is developed for investigating the stability of complex structures that consist of an assembly of stiffened rectangular panels and three-dimensional beam elements. Such panels often form one of the basic structural components of an aircraft or ship structure. In the present study, the stiffeners are treated as beam elements, and the panels between them as thin rectangular plate elements, which may be subject to membrane and/or bending and twisting actions.

The main objective of the study is the determination of the critical buckling loads and the generation of the complete force-deformation behavior of such structures within a specified load range, based on the use of a computer program developed for this purpose. The present formulation can trace through the postbuckling or post limit behavior whether it is of an ascending or descending type. A limit load extrapolation technique is automatically initiated within the computer program, when the stability analysis of an imperfect or laterally loaded structure is being carried out.

The general approach to the solution of the problem is based on the finite element method and incremental numerical solution techniques. Initially, nonlinear strain-displacement relations together with the assumed displacement functions are utilized to generate the geometric stiffness matrices for the beam and plate elements. Based on energy methods and variational principles, the basic expressions governing the behavior of the structure are then obtained. In the incremental solution process, the stiffness properties of the structure are continuously updated in order to properly account for large changes in the geometry of the structure.

The computer program developed during the course of this study is referred at as GWU-SAP, or the George Washington University Stability Analysis Program.     

Unstructured tetrahedral mesh quality analysis using an interactive haptic and visual interface

The problem of interactively probing a mesh to determine its quality is described for three-dimensional unstructured tetrahedral meshes. Mesh quality as a function of individual element error is defined for a specific class of problems. The importance of analyzing mesh quality within a geometrical representation of the mesh is discussed. The problems encountered when attempting to visualize the geometric and error information for a visually complex mesh are identified and used to motivate a design for an interactive user interface for mesh quality analysis. The primary intended user of such a system is one who is interested in per-element mesh quality, such as the developer of mesh generating software or the persons charged with generating a good quality mesh for a specific problem; however, it may also be used by end users of meshes to see the main problem areas of the mesh and to compare various available meshing strategies. The interface provides the user with necessary information about element quality in a form which allows the user to isolate “bad” mesh elements and analyze the individual contributions of element shape, orientation, geometric neighborhood, and solution behavior. The availability of this information when combined with a haptic device allows the user to easily identify poor quality mesh regions. A prototype implementation of the interface was constructed and used to examine two meshes in detail. This was done in part reflexively, to determine the feasibility of this approach to mesh quality analysis. It was also done in the interests of our larger goals, to try to determine the main contributers to poor element quality in the two meshes. User analysis of the problem meshes is presented along with visual output from the interface. A formal user study was not performed; however, informal results and timings are used to show the speed and effectiveness of the interface. Received: 15 June 2000 / Accepted: 16 May 2001 RID="*" ID="*"Current address: Cell Matrix Corporation, Blacksburg, VA, USA (http://www.cellmatrix.com) RID="**" ID="**"Research of this author is partly supported by an NSF Traineeship and the DOE-sponsored Advanced Visualization Technology Center. Communicated by G. Wittum