Graphic PIZiadas

Graphic PIZiadas

My world is in..

Categories Projectivity

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.

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: 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.

Geometría proyectiva: Construction of quadruples of points

We have seen the definition of ordered quadruples of elements, characterizing rectilinear some four points or four straight from a bundle of planes through a value or characteristic, result for the ratio of two triads determined by such elements.

We then consider the problem of obtaining, given three elements belonging to a same form of first category, series or beam, get a fourth element that determines a Tetrad of particular value.

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: Intersection of straight and tapered

Projective definition of the conical allowed to start solve classical problems of identification of new elements of the conical (new points and tangents in them), and find the intersection with a line or a tangent from an external point. These problems can be solved by various more or less complex methods and conceptually more or less laborious paths.

We will now see how to determine the two possible intersection points of a line with a taper defined by five points.

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).

Geometría proyectiva: Overlapping shapes first order

Projective overlapping shapes are a special case of projective shapes, you relate elements of the same type that share a common base.

For example, two overlapping series will have the same line as the basis of geometric shapes, two beams of the same vertex straight (concentric bundles) and two beams overlapping planes around the same axis (coaxiales).

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.