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Orientador(es)
Resumo(s)
A pr´atica do bilhar configura-se como um dom´ınio privilegiado de aplica¸c˜ao
emp´ırica de conhecimentos matem´aticos, f´ısicos e geom´etricos. A dinˆamica
do jogo permite a an´alise rigorosa de trajet´orias, ˆangulos, vetores de for¸ca
e fen´omenos de rota¸c˜ao. A mesa, delimitada por tabelas retangulares, pode
ser modelada como um plano geom´etrico no qual se formalizam princ´ıpios
como a reflex˜ao especular, a conserva¸c˜ao do momento linear e determinadas
transforma¸c˜oes sim´etricas. Assim, a execu¸c˜ao de uma tacada eficaz depende
n˜ao apenas de destreza motora, mas tamb´em da compreens˜ao (ainda que
intuitiva) de estruturas matem´aticas subjacentes.
As situa¸c˜oes de snooker, caracterizadas pela ausˆencia de linha direta entre
a bola branca e a bola-alvo devido `a interposi¸c˜ao de outras bolas, evidenciam
a necessidade de racioc´ınio geom´etrico avan¸cado. Nesses casos, o jogador
recorre a trajet´orias indiretas com uma ou m´ultiplas reflex˜oes nas tabelas.
Embora sejam tradicionalmente utilizados modelos baseados em simetrias
ou sistemas de marca¸c˜ao num´erica (como os sistemas de diamantes), tais
m´etodos revelam limita¸c˜oes em contexto competitivo.
Neste enquadramento, o presente trabalho prop˜oe uma abordagem ino
vadora, articulando fundamentos te´oricos e aplica¸c˜ao pr´atica. Introduzem-se
conceitos da teoria de grupos, em particular a¸c˜oes de grupo, bem como no¸c˜oes
de espa¸cos e transforma¸c˜oes afins. Com base nesse referencial, desenvolve-se um modelo espec´ıfico para o bilhar, orientado para a transferˆencia efetiva do
conhecimento ao contexto competitivo. O objetivo consiste em capacitar o
atleta para identificar, em situa¸c˜oes de snooker, o ponto ´otimo de contacto na
tabela, otimizando a decis˜ao e a execu¸c˜ao. Por fim, apresenta-se um c´odigo
inform´atico, concebido a partir dos resultados obtidos, destinado ao apoio ao
treino.
The practice of billiards constitutes a privileged domain for the empirical application of mathematical, physical, and geometrical knowledge. The dy namics of the game allow for rigorous analysis of trajectories, angles, force vectors, and rotational phenomena. The table, bounded by rectangular cushi ons, can be modeled as a geometric plane in which principles such as specular reflection, conservation of linear momentum, and certain symmetric trans formations are formalized. Thus, the execution of an effective shot depends not only on motor skill but also on an understanding (often intuitive) of underlying mathematical structures. Snooker situations, characterized by the absence of a direct line between the cue ball and the target ball due to the interposition of other balls, high light the need for advanced geometric reasoning. In such cases, the player re lies on indirect trajectories involving one or multiple reflections off the cushi ons. Although traditional models based on symmetries or numerical marking systems (such as diamond systems) are commonly used, these methods reveal limitations in competitive contexts. Within this framework, the present work proposes an innovative approach that articulates theoretical foundations with practical application. Concepts from group theory, particularly group actions, are introduced, along with notions of affine spaces and affine transformations. Based on this theoretical foundation, a specific model for billiards is developed, oriented toward the effective transfer of knowledge to competitive play. The objective is to enable the athlete to identify, in snooker situations, the optimal point of contact on the cushion, thereby optimizing decision-making and execution. Finally, a computer program developed from the results of this work is presented as a training support tool.
The practice of billiards constitutes a privileged domain for the empirical application of mathematical, physical, and geometrical knowledge. The dy namics of the game allow for rigorous analysis of trajectories, angles, force vectors, and rotational phenomena. The table, bounded by rectangular cushi ons, can be modeled as a geometric plane in which principles such as specular reflection, conservation of linear momentum, and certain symmetric trans formations are formalized. Thus, the execution of an effective shot depends not only on motor skill but also on an understanding (often intuitive) of underlying mathematical structures. Snooker situations, characterized by the absence of a direct line between the cue ball and the target ball due to the interposition of other balls, high light the need for advanced geometric reasoning. In such cases, the player re lies on indirect trajectories involving one or multiple reflections off the cushi ons. Although traditional models based on symmetries or numerical marking systems (such as diamond systems) are commonly used, these methods reveal limitations in competitive contexts. Within this framework, the present work proposes an innovative approach that articulates theoretical foundations with practical application. Concepts from group theory, particularly group actions, are introduced, along with notions of affine spaces and affine transformations. Based on this theoretical foundation, a specific model for billiards is developed, oriented toward the effective transfer of knowledge to competitive play. The objective is to enable the athlete to identify, in snooker situations, the optimal point of contact on the cushion, thereby optimizing decision-making and execution. Finally, a computer program developed from the results of this work is presented as a training support tool.
Descrição
Palavras-chave
Matemática Transformações afins Ações de grupo Espaços ffins Reflexão Tacada Bilhar Group actions Affine spaces Affine transformations Reflection Shot Billiards Matemática, Estatística e Aplicações . Faculdade de Ciências Exatas e da Engenharia