Projective Geometry: Autopolares triangles in Involutions in second-order series
By linking four points of a conical proyectivamente by Involutions determine the axis of involution of these proyectividades.
Given the four points needed to define an involution, We can ask ourselves How many different Involutions We can establish between them.
If we call “A” one of the points, the counterpart of this item in a particular involution can be any of the other three, being the pair of points remaining counterparts between if. We can therefore see that three different Involutions are possible as shown in Figure.
In each of these Involutions a different involution axis shall be determined.
If we get the three axes of involution on a same figure, We can obtain interesting conclusions.
If we associate as counterparts points A-A12 We will have like involution to the straight shaft E12
If we associate as counterparts points A-A23 We will have like the straight shaft e23
If we associate as counterparts points A-A31 We will have like the straight shaft E31
We see that the three axes of involution coincide with the diagonals of the full cuadrivertice determined by the homologous points of the Conic, so the polar point diagonal with respect to two of the sides of the cuadrivertice is the opposite diagonal (It contains no), as we saw when defining the Polar of a point with respect to two lines.
We see that in the triangle determined by three diagonal dots, D1, D2 and D3, each of these points is polar opposite straight. We say that this triangle is “Autopolar” with respect to the given Conic.
Geometría proyectiva : Center of involution We have seen how to determine the axis of an involution and, based on the concept of polar of a point with respect to two lines, possible Involutions which can be set from four points, with their respective shafts of involution, obtaining the associate autopolar triangle in the […]
Geometría proyectiva: Conjugate polar diameters We have seen the definition of polar conjugate diameters, given to analyze the concept of Conjugate directions:
Conjugate polar diameters: They are polar two conjugated improper point.
Let's see how we can relate this concept with the triangle's autopolar seen […]
Geometría proyectiva: Tangent from a point to a conical We have seen how to determine the points of intersection of a straight line with a Conic defined by five points. We will then see the dual problem.
This problem consists of determining the possible two straight tangent from a point to a Conic defined by five tangent.