# Geometría proyectiva: Obtaining the conical center

For the conical center will need to have poles and polar respect thereof. In particular constructions are simplified if we know tangents and contact points. We will see that is especially immediately if three tangents and their respective contact points are known, obtained from the definition of the conic by 5 data and application of the techniques disclosed to determine tangents and points of tangency.

# projective center two beams [Interactive] [Geogebra]

a conical (punctual) It is the locus of the points of intersection of two projective beams.
This model has been verified with a variational model of projective shaft made with Geogebra.

# Projective axis of two series [Interactive] [Geogebra]

Projective geometry constructions made with tools to analyze their invariants are very useful for the study of this discipline of Graphic Expression. We will see one of these constructions made with the software “GeoGebra”, in particular for determining the projective axis of two projective series.

# 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 in Involutions in second-order series.

# Geometría proyectiva: Conjugate directions

The concepts of polarity we've seen to determine the polar of a point on a line, you have allowed us to obtain the autopolar triangle of a conical setting three different involuciuones with four points, They allow us to advance in the projective definition of its notable elements, diameters, Center and axis.

One of the basics is the of “Conjugate directions”

# 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.

# 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 autopolar triangle associated which are harmonious relations of the cuadrivertice full.

In this article we will continue to enhance these elements, in particular in the autopolar triangle vertices that will determine what is known as “Center of involution”.

# Projective Geometry: Autopolares triangles in Involutions in second-order series

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.

## 25 February, 2015

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 false position method. Application of overlapping series of second order.

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.

# Geometría proyectiva: Involution in overlapping series of second order : Axis of involution

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.