For the conical center will need to have poles and polar respect thereof. In particular constructions are simplified if we know tangents and contact points. We will see that is especially immediately if three tangents and their respective contact points are known, obtained from the definition of the conic by 5 data and application of the techniques disclosed to determine tangents and points of tangency.
A conical axes are those conjugates polar diameters are orthogonal to each.
We recall that two polar conjugate diameters, necessarily pass through the center O of the conical, are the polar two points unfit (located at infinity) that they are conjugated, namely, the polar of each of these points contains the other.
These pairs of elements determine an involution of diameters (polar) Conjugates will be defined when two pairs of beams know and their homologues.
We have solved the determination of a conic defined by the two foci and focal point by the circumference of the conical.
A problem using identical concepts is determining a known conic its foci and their tangents. We will see this problem in the case of an ellipse.
One of the first problems we can solve based on the definition of conic as “locus of the centers of circumferences passing through a fixed point (focus) which are tangent to a circumference (focal circle centered the other focus)” It is the determination of the tapered from the two foci and a point.
The classic definition will be determined as the vertices A1 and A2 of the conical obtained.
We have seen that the study of conic can be made from different geometric approaches. En particular, to start analyzing conic we have defined as the ellipse locus, we said that:
Ellipse is the locus of points in a plane whose sum of distances from two fixed points, called Spotlights, It has a constant value.
This metric definition of this curve allows us to address important study relating to the tangents circumferences, known as “Problem of Apollonius” in any of its versions. When we approach the study of the parabola or hyperbola return to reframe the problem to generalize these concepts and reduce problems “fundamental problem of tangents in the case straight”, or “fundamental problem of tangents in the case circumference”, namely, determining a circumference of a “Make corradical” a tangency condition.
a conical (punctual) It is the locus of the points of intersection of two projective beams.
This model has been verified with a variational model of projective shaft made with Geogebra.
The study of conic can be made from different geometric approaches. One of the most used is the analysis that determined from planar sections in a cone of revolution.
From this definition it is possible to infer metric properties of these curves, plus new definitions of the same.
We have seen the definition of polar conjugate diameters, given to analyze the concept of Conjugate directions:
Conjugate polar diameters: They are polar two conjugated improper point.
Let's see how we can relate this concept with the triangle's autopolar seen in Involutions in second-order series.
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.
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).
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.
