Navegando por Autor "Saavedra, Beto"

Filtrar resultados informando as primeiras letras
Agora exibindo 1 - 5 de 5
  • Imagem de Miniatura
    Item
    Experiências geoanalíticas na sala de aula
    (2019-06-29) Saavedra, Beto
    Through the solution of problems in Analytic Geometry, this work aims to discuss the potential of its concepts and formulas, which are often applied mechanically. Furthermore, it seeks to show, with due modesty, that within the teaching and learning process of this discipline there are still new horizons to be discovered and explored. This work is not intended as an introduction to Analytic Geometry, but rather as a review.
  • Imagem de Miniatura
    Item
    Geometria Analítica : rotação e translação de eixos coordenados
    (2026-04-27) Saavedra, Beto
    Rotation and translation of coordinate axes are fundamental techniques in Analytic Geometry used to simplify equations. The equation xy-x+y-3=0 represents a hyperbola. The mixed term (xy) indicates that the curve is not aligned with the original coordinate axes. For this reason, we apply a rotation of the axes by an angle (\theta) in order to eliminate the mixed term and obtain a simpler form of the equation. After the rotation, a translation of the axes may also be performed, shifting the origin to the center of the curve. In this way, the canonical form of the hyperbola is obtained. The change of coordinates makes it possible to: * identify the type of conic; * find the center of the curve; * determine symmetries; * facilitate the calculation of geometric elements such as vertices, foci, and asymptotes. The video allows us to visualize dynamically the harmony between the rotated axes (x_1) and (y_1), the angle (\theta), and the equations representing the hyperbola throughout the geometric transformation.
  • Nenhuma Miniatura disponível
    Item
    Geometria Analítica: Ângulo entre Dois Planos
    (2026-11-10) Saavedra, Beto
    **Video Summary — Angle Between Two Planes** The video uses a **three-dimensional perspective and geometric animations** to explain the concept of the angle between two intersecting planes. Through the rotation of the planes, it illustrates the relationship between the angle formed by the planes and the angle between their normal vectors. The video also examines the case in which the angle between the normal vectors is obtuse, showing why its supplementary angle is considered. An algebraic derivation justifies the use of the absolute value in the formula for the cosine of the angle between two planes: **cos θ = |n⃗₁ · n⃗₂| / (‖n⃗₁‖ ‖n⃗₂‖)** where θ is the angle between the normal vectors n⃗₁ and n⃗₂ of the planes. This approach combines **three-dimensional visualization, geometric interpretation, and algebraic reasoning**, facilitating a deeper understanding of the concept.
  • Nenhuma Miniatura disponível
    Item
    Geometria analítica: equações paramétricas do plano
    (2026-05-28) Saavedra, Beto
    This material presents, in a visual and educational way, the idea of a plane in parametric form in Analytic Geometry. The plane is defined by an initial point (P_0) and by two non-parallel direction vectors, (\vec a) and (\vec b). From the equation (P = P_0 + s\vec a + t\vec b), it is shown that, as the parameters (s) and (t) vary, the points belonging to the plane are generated. Thus, the animation helps students understand how the direction vectors determine the orientation of the plane in space.
  • Imagem de Miniatura
    Item
    Uma demonstração matricial do Teorema de König (sem o uso de grafos)
    (2026-11-05) Saavedra, Beto; Kuzimoto, Diego
    This work presents a matrix-based proof of König’s Theorem without relying on graph theory, taking as its starting point the mathematical foundations of the Hungarian Method for solving optimal assignment problems. Initially, the concepts of cost matrix, optimal assignment, and equivalence between matrices are introduced, accompanied by an application illustrating the main steps of the algorithm. Subsequently, the termination of the procedure and the determination of optimal solutions are discussed. The proof of König’s Theorem is developed through a matrix-based approach, using auxiliary notation and a result referred to as **Diego’s Lemma**. This text was prepared as supporting material for the short course **MC55 – A Matrix-Based Proof of König’s Theorem Without Using Graph Theory**, delivered during the **12th Biennial Meeting of the Brazilian Mathematical Society**, held at the Federal University of Rio Grande do Norte (UFRN), in Natal, Brazil, from August 3 to 7, 2026, with a total duration of four hours. The presentation aims to provide an alternative perspective on understanding König’s Theorem and its relationship with the Hungarian Method, emphasizing elementary matrix-based arguments.