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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.

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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.

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Geometría proyectiva: Application of 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.

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Geometría proyectiva: Overlapping series of second order

When the base of a series is a conical series is second order.

As in the case of series of the first order when the overlapping series were defining, we can establish proyectividades between two sets of second order with the same base (in this case a conical).

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Geometría proyectiva: Circumference as a series of second order

A circle is a conical axes are of equal length, hence we can say that its eccentricity is zero (eccentricity = 0). We can treat the circle as one series of second order, obtained by the intersection of two beams of rays congruent counterparts (same but rotated.) This treatment will be useful to use as a projective tool and resolve the determination of double elements in overlapping concentric series and do.

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Geometría proyectiva: Determination of homologous elements in series projective

One of the first problems we must learn to work in projective geometry is the determination of homologous elements. To start the study will use the methodology to be used as usual model-based elements “points”, since it is easier to interpret. Therefore we will consider the determination of homologous elements in series projective:
Given two projective series defined by three pairs of elements (points) counterparts, determine the counterpart of a given point.

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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.

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Geometría proyectiva: Projective projective axis of two series

The operational prospects relationships is reduced to the concepts of belonging, so we will use these techniques to suit projective models simplify obtaining homologous elements.
How can we define two projective series? On how many homologous elements are necessary to determine a projectivity?How can we obtain homologous elements?

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