# 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.

# Geometría proyectiva: Definition of the conical projective

Conic curves, further treatment of the metric based on the notions of tangency, have a projective treatment that relies on the concepts of sets and projective bundles.

We will see two definitions of conic adapted to “World points” o al “world of straight” according to the interest, in what is defined as the definitions “point” or “tangential” of conic curves.

# 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.

# 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?

# Geometría proyectiva: Projectivity

The relationship called “cuaterna” or “double ratio of four elements” to define the general homographic transformations perspectivity and projectivity.

# Geometría proyectiva: Perspectivity

Projective foundations are based on the definitions of "ordered triples of elements" and “quaternions for defining the cross ratio”, and relationships called “perspectives” between elements of identical or different nature.
These perspectives relations, that will be used in determining projections representation systems, defined from two projective operators:
Projection
Section