The concept of polarity is linked to the harmonic separation.
This concept is Basic for the determination of the fundamental elements of conics, as its Center, conjugate diameters, axes ….
It will allow to establish new transformations which include homographies and correlations of great importance.
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.
Application “Geogebra” It allows you to develop dynamic constructions in which we can modify the position of the elements forming it, keeping the geometric constraints of these figures, allowing the invariants of the same show. This tool can be a valuable aid for students.
Professor Juan Alonso Alriols collaborated in the introduction of this tool in the teachings of “Graphic Expression” at the Polytechnic University of Madrid, providing examples of high interest. You can see an example of his work in the “Dynamic construction of double reason for four points” accompanying this entry, that has added a driver text for use in our classes.
We have seen the definition of ordered quadruples of elements, characterizing rectilinear some four points or four straight from a bundle of planes through a value or characteristic, result for the ratio of two triads determined by such elements.
We then consider the problem of obtaining, given three elements belonging to a same form of first category, series or beam, get a fourth element that determines a Tetrad of particular value.
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.
