Connecting four points of a conical proyectivamente by Involutions we determine the axis of involution of these proyectividades.
Given the four points needed to define an involution, We can ask many different Involutions can establish between them.
One of the most used in projective geometry geometric figures is the of the “Full Cuadrivertice”, or its dual “Full ring”.
Generally, a cuadrivertice is formed by four points, so on the plane this figure has 8 degree of freedom (2 coordinates for each vertex) and they will be needed 8 restrictions to determine one concrete.
The theoretical models of projective geometry can be proposing problems that are not of direct application. We will have that “dress up” therefore exercises to infer in the student further analysis and a transverse treatment of knowledge: Can I apply what they learn to solve this problem?.
After analyzing in detail the operations with overlapping series of second order, Let's see an example of application which does not consist in obtaining new tangents or points of contact of a conical.
Involutionary transformations are applications bijective of great interest to be applied in geometric constructions, since they simplify them considerably.
We will see how defined an involution in second-order series, with base a conical, Comparing the new model of transformation with overlapping series of second order previously studied.
In geometry, we speak often with terms that, in some cases, they are not sufficiently important in everyday language. This leads to create barriers in the interpretation of some simple concepts.
One of the terms that I have been asked several times in class is the of “Involution”. We define the involution.
What is an involution?
You do projective concepts that we have developed to studying overlapping of second order, whose base is a conical, They allow to solve problems of determination of points of contact in the tangents of a Conic defined by five tangent or five restrictions through the combination of tangent and their respective tangent points. We will see the implementation of Brianchon point in this type of problems
To study the tangential Conic, and in particular the proyectividades between beams of second order superimposed on a same curve, We can rely on the dual study of the accomplished with overlapping series of second order.
The projective concepts that we have developed to study the overlapping series of second order, whose base is a conical, They allow to solve problems of determination of tangent points of a Conic defined by five points or five restrictions through the combination of points and tangents with their respective points of tangency.
Vamos a resolver un sencillo problema planteado anteriormente en el que deberemos determinar un lugar geométrico básico para la determinación de su solución, un problema en el que hay que encontrar un punto del plano que cumpla unas condiciones geométricas dadas.
La intersección de dos lugares geométricos planos nos determinará un número finito de puntos que serán las posibles soluciones del problema.
Las técnicas de solución de problemas basadas en la intersección de lugares geométricas se suelen asociar a problemas sencillos de la geometría clásica.
En estos casos es el planteamiento de la solución lo que entraña la mayor complejidad, ya que los lugares geométricos derivados suelen ser elementos geométricos sencillos.
Determinar un punto P desde el que se observe bajo el mismo ángulo a los tres lados de un triángulo ABC.
Vamos a resolver el problema de determinar un cuadrado, cuyos vértices se encuentran sobre elementos geométricos dados.
En particular fijaremos los correspondientes a una de sus diagonales sobre una recta, otro de los vértices en una recta diferente y el cuarto vértice sobre una circunferencia.
