# Metric geometry : Investment beam circumferences

Transformation through investment in geometric shapes grouped elements can be of interest to use the investment as a tool for analysis in complex problems. In this case study transforming “beams circumferences corradicales” through various investments that transform. Later these transformations need to solve the problem “Apolonio” (circumference with three tangency constraints) o la “Generalization of the problem of Apollonius” (circumferences with three angular restrictions).

# 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: Application of second-order overlapping beams

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

# Geometría proyectiva: You do overlapping of second order

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.

# Geometría proyectiva: Determination of homologous elements in projective beams

One of the first problems we must learn to work in projective geometry is the determination of homologous elements, both in series and in bundles and in any provision of bases, or separate superimposed.

To continue the study of the methodology to be used will use the dual model the elements based on “points”, ie with straight, further assuming that the bases of the respective beams are separated relate.

# Geometría proyectiva: Projective center of two projective bundles

Using the laws of duality in projective models can get a set of properties and dual theorems from other previously deducted. Obtaining homologous elements in the projective case series was performed by obtaining intermediate pespectividades allowed perspectival do we get what we have called “projective axis”. We will see that in the case of projective bundles, Dual reasoning leads us to determine projective centers.

# Metric geometry : Generalization of the fundamental problem of tangents :

We have solved the fundamental problem we have called for tangents when presented with tangency conditions on a circle or a straight. Conceptually we can assume that both problems are the same, if we consider the straight as a circle of infinite radius. The statement therefore posed circumferences obtaining through two points were tangent to a straight or tangent to a circle.

# Categories projective geometric shapes and operations

Geometric shapes are categorized.
From a viewpoint parametric, the category of a geometric shape is the number of variables or data necessary for referencing an element thereof.