A completion of hypotheses method for 3D-geometry. 3D-extensions of Ceva and Menelaus theorems

A method that automates hypotheses completion in 3D-Geometry is presented. It consists of three processes: defi ning the geometric objects in the confi guration; determining the hypothesis conditions of the confi guration (through a point-on-object declaration method); and applying an algebraic automatic theorem proving method to obtain and prove the sufficiency of complementary hypothesis conditions. To avoid as much as possible the appearance of rational expressions, projective coordinates are used (although affine and Euclidean problems can also be treated). A Maple implementation of the method has been used to extend to 3D classic 2D geometric theorems like Ceva's and Menelaus'

A brief note on the approach to the conic sections of a right circular cone from dynamic geometry

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 implicitequation of the conic

Connecting the 3D DGS Calques3D with the CAS Maple

Many (2D) Dynamic Geometry Systems (DGSs) are able to export numeric coordinates and equations with numeric coefficients to Computer Algebra Systems (CASs). Moreover, different approaches and systems that link (2D) DGSs with CASs, so that symbolic coordinates and equations with symbolic coefficients can be exported from the DGS to the CAS, already exist. Although the 3D DGS Calques3D can export numeric coordinates and equations with numeric coefficients to Maple and Mathematica, it cannot export symbolic coordinates and equations with symbolic coefficients. A connection between the 3D DGS Calques3D and the CAS Maple, that can handle symbolic coordinates and equations with symbolic coefficients, is presented here. Its main interest is to provide a convenient time-saving way to explore problems and directly obtain both algebraic and numeric data when dealing with a 3D extension of "ruler and compass geometry". This link has not only educational purposes but mathematical ones, like mechanical theorem proving in geometry, geometric discovery (hypotheses completion), geometric loci finding... As far as we know, there is no comparable "symbolic" link in the 3D case, except the prototype 3D-LD (restricted to determining algebraic surfaces as geometric loci)

A fast functional approach to personalized menus generation using set operations

The authors developed some time ago a RBES devoted to preparing personalized menus at restaurants according to the allergies, religious constraints, likes and other diet requirements as well as products availability. A first version was presented at the "Applications of Computer Algebra 2015" (ACA'2015) conference and an improved version to the "5th European Seminar on Computing" (ESCO2016). Preparing personalized menus can be specially important when traveling abroad and facing unknown dishes in a menu. Some restaurants include icons in their menu regarding their adequateness for celiacs or vegetarians and vegans, but this is not always a complete information, as it doesn't consider, for instance, personal dislikes or uncommon allergies. The tool previously developed can obtain, using logic deduction, a personalized menu for each customer, according to the precise recipes of the restaurant and taking into account the data given by the customer and the ingredients out of stock (if any). Now a new approach has been followed, using functions and set operations and the speed has been increased by three orders of magnitude, allowing to deal with huge menus instantly. Both approaches have been implemented on the computer algebra system Maple and are exemplified using the same recipes in order to compare their performances.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Computer Algebra-based RBES personalized menu generator

People have many constraints concerning the food they eat. These constraints can be based on religious believes, be due to food allergies or to illnesses, or can be derived just from personal preferences. Therefore, preparing menus at hospitals and restaurants can be really complex. Another special situation arise when travel- ing abroad. It is not always enough to know the brief description in the restaurant menu or the explanation of the waiter. For example, “calamares en su tinta” (squid in its own ink) is a delicious typical Spanish dish, not well-known abroad. Its brief description would be “squid with boiled rice in its own (black) ink”. But an in- gredient (included in a small amount, in order to thicken the sauce) is flour, a fact very important for someone suffering from celiac disease. Therefore, we have con- sidered that it would be very interesting to develop a Rule Based Expert System (RBES) to address these problems. The rules derive directly from the recipes and contain the information about required ingredients and names of the dishes. We distinguish: ingredients and ways of cooking, intermediate products (like “mayon- naise”, that doesn’t always appear explicitly in the restaurants’ menus) and final products (like “seafood cocktail”, that are the dishes listed in the restaurant menu). For each customer at a certain moment, the input to the system are: on one hand, the stock of ingredients at that moment, and on the other, the religion, allergies and restrictions due to illnesses or personal preferences of the customer. The RBES then constructs a “personalized restaurant menu” using set operations and knowl- edge extraction (thanks to an algebraic Groebner bases-based inference engine[1]). The RBES has been implemented in the computer algebra system Maple TM 18(us-ing its convenient Embedded Components) and can be run from computers and tablets using Maple TM or the Maple TM PlayerUniversidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

A RBES for Generating Automatically Personalized Menus

Food bought at supermarkets in, for instance, North America or the European Union, give comprehensive information about ingredients and allergens. Meanwhile, the menus of restaurants are usually incomplete and cannot be normally completed by the waiter. This is specially important when traveling to countries with a di erent culture. A curious example is "calamares en su tinta" (squid in its own ink), a common dish in Spain. Its brief description would be "squid with boiled rice in its own (black) ink", but an ingredient of its sauce is flour, a fact very important for celiacs. There are constraints based on religious believes, due to food allergies or to illnesses, while others just derive from personal preferences. Another complicated situation arise in hospitals, where the doctors' nutritional recommendations have to be added to the patient's usual constraints. We have therefore designed and developed a Rule Based Expert System (RBES) that can address these problems. The rules derive directly from the recipes of the di fferent dishes and contain the information about the required ingredients and ways of cooking. In fact, we distinguish: ingredients and ways of cooking, intermediate products (like sauces, that aren't always made explicit) and final products (the dishes listed in the menu of the restaurant). For a certain restaurant, customer and instant, the input to the RBES are: actualized stock of ingredients and personal characteristics of that customer. The RBES then prepares a "personalized menu" using set operations and knowledge extraction (thanks to an algebraic inference engine [1]). The RBES has been implemented in the computer algebra system MapleTM2015. A rst version of this work was presented at "Applications of Computer Algebra 2015" (ACA'2015) conference. The corresponding abstract is available at [2].Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Automatización e implementación de algunos problemas algebráicos y geométricos

En esta tesis doctoral se estudian varios problemas geométricos clásicos: 1. Automatización e implementación del grupo equiforme de transformaciones geometncas de R 2 2 2. Simulación de los 17 grupos de simetría de R , a través de la generación constructiva de mosaicos periódicos. 3. Simulación de la transformación geométrica "Inversión" y su aplicación a la automatización e implementación del problema de Apolonio (de la circunferencia tangente a otras tres dadas). 4. Aplicación de métodos algebraicos (método de las bases de Groebner y método de seudodivisiones de Wu) a la automatización de un criterio de no-ramificación de ideales primos en extensiones algebraicas y a la demostración automática de un teorema original de Geometría Sintética. El objetivo de los tres primeros es efectuar una adaptación de estos problemas geométricos, construyendo algoritmos apropiados que permitan automatizar su utilización, llegando a su implementación. Y el objetivo del cuarto, es aplicar métodos algebraicos de demostración automática a probar un teorema geométrico de penosa demostración por métodos tradicionales y a automatizar la aplicación de un criterio original de no-ramificación. La naturaleza de los tres primeros problemas induce a utilizar técnicas de Geometría Vectorial, lo que justifica que la implementación convenga realizarla en un lenguaje que disponga de la denominada "Geometría de la Tortuga". Como ninguna de las implantaciones usuales posee todos los requerimientos deseados, se ha desarrollado una adaptación apropiada denominada "Turtgeom", que permite mejorar notablemente la graficación de las simulaciones citadas. Para el cuarto problema se ha de utilizar algún sistema de cómputo algebraico que contenga implementaciones de "bases de Groebner", habiendo optado por los sistemas Maple y Reduce. El trabajo desarrollado en ella se inició con el Proyecto de Investigación "Simulación Informática de Problemas Algebraicos y Geométricos" (UCP40/87). A B S T R A C T In this Thesis we deal with several classic geometric problems: 1. Implementation of the Equiform Group of geometric transformations in IR . 2. Simulation of the 17 Wallpaper Patterns, with constructive generation of periodic mosaics. 3. Simulation of the geometric transformation "Inversión" and its application to the implementation of the Apollonius problem (to determine the tangent circumferences to three other ones given). 4. Application of algebraic methods (Groebner's basis and Wu's pseudoremainder) to an algebraic problem and a geometric problem. The algebraic problem consists of verifying a criterium of no-ramification of prime ideáis in algebraic extensions and the geometric one consists of the automatic proving of an original theorem of Synthetic Geometry. The main objective of the first three problems is to do an adaptation of these geometric problems, building the appropriate algorithms that will allow us to implementat the former. The objective of the fourth problem is applying algebraic methods of automatic theorem proving in order to prove a geometric theorem that is really time-consuming using traditional methods. The application of an original criterium of no-ramification is also implemented. The nature of the first three problems lead us to use techniques of Vectorial Geometry. Something that justifies the use of a language that includes the so called "Turtle Geometry". As none of the usual implementations has all the necessary requirements, an appropriate implementation called "Turtgeom" has been developed. It allows us to improve the graphics of the mentioned simulation remarkably. For the fourth problem a Computer Algebra System that includes an implementation of Groebner Basis should be used. Therefore, we have chosen the MAPLE and REDUCE systems. The work developed in this Thesis was begun with a Research Project entitled "Computer Simulation of Algebraic and Geometric Problems" (UCP 40/87)

An Approach to Overtaking Station Layout Diagram Design Using Graphs

The authors have approached in the past different railway engineering problems (e.g. [1,2]) and have developed software for the Spanish Railway Foundation [3], mainly using computer algebra systems (CAS). Now a CAS is used for designing and implementing a software package that allows to compare different layout diagrams for overtaking stations [4], from the point of view of its flexibility (for instance in degraded working conditions). The most common cases of overtaking stations (stations on double track lines with one or two sidings each side of the main line and one or two crossovers at the two throats of the overtaking station) are considered as illustration. It will be shown how, surprisingly, the usual position of the crossovers is not optimum from this point of view. The key idea is to use graph theory to determine the number of pairs of non conflicting itineraries (one in each direction) that can be simultaneously authorized by the railway interlocking system. The package can be applied to an overtaking station with any layout diagram. This is an important issue: for instance, the Spanish infrastructure administrator (Adif) is now planning a new track layout for Madrid Chamartín station [5] and the Iberian gauge part of Madrid Atocha station.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

A Parametric Approach to 3D Dynamic Geometry

Dynamic geometry systems are computer applications allowing the exact on-screen 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, using numerical data to manage geometric knowledge and elementary methods to compute 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, thus enabling a full algebraic knowledge about objects such as loci and envelopes. This parametric approach is also a prerequisite for performing automatic proof. Basic functions are defined for symbolically checking the truth of statements. Using recent results 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, which, however, lacks specific functions for finding loci and envelopes)