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
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.
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.
Application “Geogebra” It allows you to develop dynamic constructions in which we can modify the position of the elements forming it, keeping the geometric constraints of these figures, allowing the invariants of the same show. This tool can be a valuable aid for students.
Professor Juan Alonso Alriols collaborated in the introduction of this tool in the teachings of “Graphic Expression” at the Polytechnic University of Madrid, providing examples of high interest. You can see an example of his work in the “Dynamic construction of double reason for four points” accompanying this entry, that has added a driver text for use in our classes.
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.
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.
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.
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).
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.
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.
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.
