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.
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.
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.
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?
The relationship called “cuaterna” or “double ratio of four elements” to define the general homographic transformations perspectivity and projectivity.
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
Among the most important curves are studied in geometry is called “Conic curves”. Another common name for these curves is the “Conic Sections” because the first definition given for them, by Apollonius of Perge, was from the sections in a cone of revolution.
One of the most geometric games there is the “Billiard game”, in which using a drum with a wad (a pool cue) on a ball, we must ensure that this impact on one or more other arranged in a rectangular table. With the “The taco de bill” effects can be given to balls, but if you just hit them in the center, behavior can be compared to the classical transformations that are studied in the axial symmetries.
Let the solution to the problem proposed arc capable application, that we proposed with the following statement:
Determine two lines that are based on a point P outside a line r, an angle formed between "alpha" and cut given to the line as a segment of length "L".
The arc geometry applications capable of an angle on a given segment are many and varied:
From the proof of a theorem, the intermediate solution of a problem or direct application in a case, We can see this construction repeatedly widespread.
One of the most comprehensive articles they have written my students in geometry classes is describing how to solve the so-called “Apollonius problems”.
Determining come straight circumferences or geometric constraints defined by the tangents are based on a family of geometric problems of great interest.
