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Automatic determination of envelopes and other derived curves within a graphic environment
, 2004
"... 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 th ..."
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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.
Automated Generation of Kempe Linkage and its Complexity
"... Abstract. It is a famous result of Kempe that we may design a linkage to generate any given plane algebraic curve. In this paper, Kempe’s result is improved to give a precise algorithm for generating Kempe’s linkage.. We also proved that for an algebraic plane curve of degree n, Kempe’s linkage uses ..."
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Abstract. It is a famous result of Kempe that we may design a linkage to generate any given plane algebraic curve. In this paper, Kempe’s result is improved to give a precise algorithm for generating Kempe’s linkage.. We also proved that for an algebraic plane curve of degree n, Kempe’s linkage uses at most O(n 4) links. We also report our efforts to implement a program which may generate the Kempe’s linkage and simulate the generation process of the plane curves. 1.
On the Parametric Representation of Dynamic Geometry Constructions
"... Abstract. This paper describes an ongoing implementation of an open source library dealing with parametric representation of dynamic geometry constructions. We show how some current issues in standard dynamic geometry environments, such as computing envelopes of lines which are not primitive objects ..."
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Abstract. This paper describes an ongoing implementation of an open source library dealing with parametric representation of dynamic geometry constructions. We show how some current issues in standard dynamic geometry environments, such as computing envelopes of lines which are not primitive objects known by the geometric system, can be efficiently solved using the correspondence between geometry and algebra. We also propose enriching the tool (or macro) mechanism, available in some environments, with our parametric approach. Finally, some problems arising from the algebraic method considered are also studied.
Geometric Constraint Solving with Linkages and Geometric Method for Polynomial
"... Abstract. In this paper, we introduce linkages as new tools and show that this tool is complete in certain sense, i.e., any diagram that can be described constructively can be drawn with linkages. As an application, we show that the simplest constrained graph which is beyond the scope of Owen and Ho ..."
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Abstract. In this paper, we introduce linkages as new tools and show that this tool is complete in certain sense, i.e., any diagram that can be described constructively can be drawn with linkages. As an application, we show that the simplest constrained graph which is beyond the scope of Owen and Hoffmann’s triangle decomposition methods can be transformed to pure geometric constructive form. To solve the equations raised from linkage constructions, we propose a geometric method which is based on dynamic locus generation. 1.
Elaine Kant The Diagrammatic Specification and Automatic Generation of Geometry Subroutines
, 2010
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Some issues on the automatic computation of plane envelopes in interactive environments
"... This paper addresses some concerns, and describes some proposals, on the ellusive concept of envelope of an algebraic family of varieties, and on its automatic computation. We describe how to use the recently developed Grobner Cover algorithm to study envelopes of families of algebraic curves, and w ..."
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This paper addresses some concerns, and describes some proposals, on the ellusive concept of envelope of an algebraic family of varieties, and on its automatic computation. We describe how to use the recently developed Grobner Cover algorithm to study envelopes of families of algebraic curves, and we give a protocol towards its implementation in dynamic geometry environments. The proposal is illustrated through some examples. A beta version of GeoGebra is used to highlight new envelope abilities in interactive environments, and limitations of our approach are discussed, since the computations are performed in an algebraically closed eld.
The Investigation of Prospective Mathematics Teachers’ Solutions of A Locus Problem
"... The purpose of this study is to investigate problem solving processes of prospective elementary mathematics teachers on locus problems by using paperpencil and Dynamic Geometry Software (DGS). In the research designed as a case study the partipicants were 19 senior prospective teachers. This data w ..."
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The purpose of this study is to investigate problem solving processes of prospective elementary mathematics teachers on locus problems by using paperpencil and Dynamic Geometry Software (DGS). In the research designed as a case study the partipicants were 19 senior prospective teachers. This data was collected by using diary notes and worksheets of the prospective teachers. The qualitative data were analyzed by following the stages suggested by the Miles and Huberman (1994, p.10): data reduction, data display and conclusion drawing/verification. In the research it was determined that prospective teachers experienced various situations such as diffuculty in understanding the question, guessing the answer, and making generalization in paperpencil application. Contrary to paperpencil solutions in the process of the software solution, prospective teachers got opportunity to try different solutions, to make hypothesis, to test their hypothesis and to make generalization by using the software and thus they reached expected solutions.