Connecting the 3D DGS Calques3D with the CAS Maple

Roanes–Lozano, Eugenio; Labeke, Nicolas Van; Roanes-Macı́as, Eugenio

doi:10.1016/j.matcom.2009.09.008

Cited by 7 publications

(7 citation statements)

References 24 publications

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“…Few authors discussed automated theorem proving in 3D geometry, see for instance [4,20,21]. Their approaches are mainly based on algebra and, thus they have to translate geometric statements into algebraic systems and to choose appropriate coordinates rather than using an axiomatic approach.…”

Section: Discussionmentioning

confidence: 99%

Two New Ways to Formally Prove Dandelin-Gallucci's Theorem

Braun

Magaud

Schreck

2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

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Mechanizing proofs of geometric theorems in 3D is significantly more challenging than in 2D. As a first noteworthy case study, we consider an iconic theorem of 3D geometry: Dandelin-Gallucci's theorem. We work in the very simple but powerful framework of projective incidence geometry, where only incidence relationships are considered. We study and compare two new and very different approaches to prove this theorem. First, we propose a new proof based on the well-known Wu's method. Second, we use an original method based on matroid theory to generate a proof script which is then checked by the Coq proof assistant. For each method, we point out which parts of the proof we manage to carry out automatically and which parts are more difficult to automate and require human interaction. We hope these first developments will lead to formally proving more 3D theorems automatically and that it will be used to formally verify some key properties of computational geometry algorithms in 3D. CCS CONCEPTS• Theory of computation → Automated reasoning; • Computing methodologies → Theorem proving algorithms.

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Section: Discussionmentioning

confidence: 99%

Two New Ways to Formally Prove Dandelin-Gallucci's Theorem

Braun

Magaud

Schreck

2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

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“…In [22] the authors illustrate their proposal by computing an ellipsoid as a locus using a natural extension of the gardener's method. In our setting, and using their element names and values, the construction is FreePoint('Pt1',0,0,0); FreePoint('Pt2',1,0,0) FreePoint('Pt3',0,2,0); Line('L13','Pt1','Pt3') PointOnObject('Pt5','L13'); Sphere('Sph8','Pt1','Pt1','Pt5') Sphere('Sph9','Pt2','Pt3','Pt5') IntersectionObjectObject('Cr10','Sph8','Sph9') They state that "paramGeo3D has automatically obtained as locus (without the user having to type anything)" when, asking for Cr10, the system returns…”

Section: Loci Functionsmentioning

confidence: 99%

“…Botana and Valcarce [1,3] develop a DG environment that communicates with CoCoA [10] and Mathematica, and where Gröbner bases are used for loci finding and automated proof and discovery. Botana has also implemented a Mathematica based web application [4] for remotely dealing with 3D geometric constructions, and Roanes-Lozano et al [22] have replied this approach using Maple with local access only.…”

Section: Introductionmentioning

confidence: 99%

A parametric approach to 3D dynamic geometry

Botana

2014

Mathematics and Computers in Simulation

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Dynamic geometry systems are computer applications allowing the exact onscreen drawing of geometric diagrams and their interactive manipulation by mouse dragging. Whereas there exists an extensive list of 2D dynamic geometry environments, the number of 3D systems is reduced. Most of them, both in 2D and 3D, share a common approach for the management of geometric constructions, using numerical data for the present diagram instance and elementary methods for computing derived objects. This paper deals with a parametric approach for automatic management of 3D Euclidean constructions. An open source library, implementing the core functions in a 3D dynamic geometry system, is described here. The library deals with constructions by using symbolic parameters, so allowing a full algebraic knowledge about objects such as loci and envelopes. This parametric approach is also a prerequisite for performing automatic proof, and basic functions are defined for symbolically checking the truth of statements. Using recent results coming from the theory of parametric polynomial systems solving, the bottleneck in the automatic determination of geometric loci and envelopes is solved. As far as we know, there is no comparable library in the 3D case, except the paramGeo3D library (designed for computing equations of simple 3D geometric objects, but it lacks specific functions for finding loci and envelopes).

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“…Nowadays there are different powerful 3D dynamic geometry systems (DGS) such as GeoGebra 5 [2], Calques 3D [7,10] and Cabri Geometry 3D [11].…”

Section: Introductionmentioning

confidence: 99%

A Brief Note on the Approach to the Conic Sections of a Right Circular Cone from Dynamic Geometry

Roanes–Lozano

2017

Math.Comput.Sci.

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Abstract. Nowadays there are different powerful 3D dynamic geometry systems (DGS) such as GeoGebra 5, Calques 3D and Cabri Geometry 3D. An obvious application of this software that has been addressed by several authors is obtaining the conic sections of a right circular cone: the dynamic capabilities of 3D DGS allows to slowly vary the angle of the plane w.r.t. the axis of the cone, thus obtaining the different types of conics. In all the approaches we have found, a cone is firstly constructed and it is cut through variable planes. We propose to perform the construction the other way round: the plane is fixed (in fact it is a very convenient plane: z = 0) and the cone is the moving object. This way the conic is expressed as a function of x and y (instead of as a function of x, y and z). Moreover, if the 3D DGS has algebraic capabilities, it is possible to obtain the implicit equation of the conic.

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Connecting the 3D DGS Calques3D with the CAS Maple

Cited by 7 publications

References 24 publications

Two New Ways to Formally Prove Dandelin-Gallucci's Theorem

Two New Ways to Formally Prove Dandelin-Gallucci's Theorem

A parametric approach to 3D dynamic geometry

A Brief Note on the Approach to the Conic Sections of a Right Circular Cone from Dynamic Geometry

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