# Reversing a point. 10 constructions for obtaining [I- Metrics]

One recommendation I always do my students is to try to solve the same problem in different ways, instead of many times the same problems with almost similar statements.

We see a problem with metric or projective approaches in each case.

In one of my last classes we propose are obtaining the inverse of a point, an investment in the center and power is known. The proposed statement was as follows:

Since the square in Figure, in which one vertex is the center of inversion and the opposite vertex is a double point, determining the inverse of the point A (adjacent vertex).

# Geometría proyectiva: Obtaining conical shafts from two pairs Diameters Polar Conjugates

A conical axes are those conjugates polar diameters are orthogonal to each.

We recall that two polar conjugate diameters, necessarily pass through the center O of the conical, are the polar two points unfit (located at infinity) that they are conjugated, namely, the polar of each of these points contains the other.

These pairs of elements determine an involution of diameters (polar) Conjugates will be defined when two pairs of beams know and their homologues.

# 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.

# To be Professor of drawing in high school you need a Master

To become Professor of technical drawing in secondary, What to do?

Many of my students have asked me what to do to be Professor of drawing, course that I teach at the University. The answer is always the same do teacher what? It is not the same be University professor who became an Institute Professor.

# 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.

# Projective Geometry: Full Cuadrivertice

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.