APOLONIO DE PERGA Trabajos Secciones cónicas. hipótesis de las órbitas excéntricas o teoría de los epiciclos. Propuso y resolvió el. Nació Alrededor Del Apolonio de Perga. Uploaded by Eric Watson . El libro número 8 de “Secciones Cónicas” está perdido, mientras que los libros del 5. In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the Greek mathematicians with this work culminating around BC, when Apollonius of Perga undertook a systematic study of their properties.

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Apolonio de Pergamo by cristobal arenas on Prezi

Thus, a polarity relates a point Q with a line q and, following Gergonneq is called the polar of Q and Q the pole of q. Apollonius’s work was translated into Arabic and much of his work only survives through the Arabic version.

One way to do this is to introduce homogeneous coordinates and define a conic to be the set of points whose coordinates satisfy an irreducible quadratic equation in three variables or equivalently, the zeros of an irreducible quadratic form. For instance, given a line containing the points A and Bthe midpoint of line segment AB is defined as the point C which is the projective harmonic conjugate of the point of intersection of AB and the absolute line, with respect to A and Apolonik.

However, as the point of intersection is the apex of the cone, the cone itself degenerates to a cylinderi. Birkhoff—Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves. An ellipse and a hyperbola each have two foci and distinct directrices for each of them.

By the Principle of Duality in a projective plane, the dual of each point is a line, and the dual of a locus of points a set of points satisfying some condition is called conicad envelope of lines.

It has been mentioned that circles in the Euclidean plane can not be defined by the focus-directrix property. Unless otherwise stated, “conic” in this article will refer to a non-degenerate conic.

These three needs can be related in some way to the broad subdivision of mathematics into the study of structure, space and change. The three types of conic apolpnio will reappear in the affine plane obtained by choosing a line of the projective space to be the line at infinity. The Aplonio plane R 2 is embedded in the real projective plane by adjoining a line at infinity and its corresponding points at infinity so that all the lines of a parallel class meet on this line.

Conic section

If a conic section has one real and one imaginary point at infinity, or two imaginary points that are not conjugated then it is not a real conic section, because its coefficients cannot be real. In analytic geometrya conic may be defined as a plane algebraic curve of degree 2; that is, as the set of points whose coordinates satisfy a quadratic equation in two variables. In all cases, a and b are positive. The three types of conic section are the hyperbolathe parabolaand the ellipse.


In particular two conics may possess none, two or four possibly coincident intersection points. Analytic theory Elliptic function Elliptic integral Fundamental pair of periods Modular form.

The latus rectum is the chord parallel to the directrix and passing through the focus or one of the two foci. If a conic in the Euclidean plane is being defined by the zeros of a quadratic equation that is, as a quadricthen the degenerate conics are: All mirrors in the shape of a non-degenerate conic section reflect light coming from or going toward one focus toward or away from the other focus.

A special case of the hyperbola occurs when its asymptotes are perpendicular. A non-degenerate conic is completely determined by five points in general position no three collinear in a plane and the system of conics which pass through a fixed set of four points again in a plane and no three collinear is called a pencil of conics. The first four of these forms are symmetric about both the x -axis and y -axis for the circle, ellipse and hyperbolaor about the x -axis only for the parabola.

The eccentricity of an ellipse can be seen as a measure of how far the ellipse deviates from being circular. The conic sections have some very similar properties in the Euclidean plane and the reasons for this become clearer when the conics are viewed from the perspective of a larger geometry. Several metrical concepts can be defined with reference to these choices. The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone — for a right cone, see diagram, this means that the cutting plane is perpendicular to the symmetry axis of the cone.

However, there are several methods that are secciones to construct as many individual points on a conic, with straightedge and compass, as desired. The classification mostly arises due to the presence of a quadratic form in two variables this corresponds to the associated discriminantbut can also correspond to eccentricity.

Then Archimedes and Apollonius appears. The eccentricity of a circle is defined to be zero and its focus is the center of the circle, but there is no line in the Euclidean plane that is its directrix. These 5 items 2 points, 3 lines uniquely determine the conic section. In other projects Wikimedia Commons. The classification into elliptic, parabolic, and hyperbolic is pervasive in mathematics, and often divides a field into sharply distinct subfields. The two lines case occurs when the quadratic expression factors into two linear factors, the zeros of each giving a line.


Teubner, Leipzig from Google Books: There are some authors who define a conic as a xonicas nondegenerate quadric. After introducing Cartesian coordinates the focus-directrix property can be used to produce equations that the coordinates of the points of the conic section must satisfy.

If the angle is acute then the conic is an ellipse; if the angle is right then the conic is a parabola; and if the angle is obtuse then the conic is a hyperbola but only coonicas branch of the curve. This can be done for arbitrary projective planesbut to obtain the real projective apolonii as the extended Euclidean plane, some specific choices have to be made. The reflective properties of the conic sections are used in the design of searchlights, radio-telescopes and some optical telescopes.

In the case of an ellipse the squares of the two semi-axes are given by the denominators in the canonical form. Furthermore, the four base points determine three line pairs degenerate conics through the base points, each line of the pair containing exactly two base points and so each pencil of conics will contain at most three degenerate conics.

Thus there is a 2-way classification: The association of lines of the pencils can be extended to obtain other eprga on the ellipse.

The solutions to a system of two second degree equations in two variables may be viewed as the coordinates of the points of intersection of two generic conic sections.

The linear eccentricity c is the distance between the center and the focus or one of the two foci. The circle is a special kind of ellipse, although historically it had been considered as a fourth type as it was by Apollonius. Elliptic function Elliptic integral Fundamental pair of periods Modular form.

Another method, based on Steiner’s construction and which is useful in engineering applications, is the parallelogram methodwhere a conic is constructed point by point by means of connecting certain equally spaced points on a horizontal line and a vertical line. A point on no seccines line is said to be an interior point or inner point of seccilnes conic, while a point on two tangent lines is an exterior point or outer point.