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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Formally, given any five points in the plane in general linear positionmeaning no conics collinearthere is a unique conic passing through them, which will be non-degenerate; this is true in both the Euclidean plane and its extension, the real projective plane.

Any point in the plane is on either zero, one or two tangent lines of a conic.

The latus rectum is the chord parallel to the directrix and passing through the focus or one of the two foci. In mathematicsa conic section or simply conic is a curve obtained as the intersection of the surface of a cone with a plane.

A special case of the hyperbola occurs when its asymptotes are perpendicular. A conic in a projective plane that contains the two absolute points is called a circle.

Hero of Alexandria is credited with Heron’s formula for calculating the area of a scalene triangle and be the first to recognize the possibility that negative numbers have square roots; also significant progress were made in applied mathematics as trigonometry. Yet another general method uses the polarity property to construct the tangent envelope of a conic a line conic.

Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically.

Because every straight line intersects a conic section twice, each conic section has two points at infinity the intersection points with the line at infinity. There are some authors who define a conic as a two-dimensional nondegenerate quadric. In homogeneous coordinates a conic section can be represented as:. More technically, the set of points that are zeros of a quadratic form in any number of variables is called a quadricand the irreducible quadrics in a two dimensional projective space that is, having three variables are traditionally called conics.

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Treatise on conic sections

Euclid was an important representative in the history of mathematics with his work The treaty of the elements, which influenced the development of geometry. Recall that the principal axis is the line joining the foci of an ellipse or hyperbola, and the center in these cases is the midpoint of the line segment joining the foci. That is, there is a projective transformation that will map any non-degenerate conic to any other non-degenerate conic. In all cases, a and b are positive.

Retrieved from ” https: It has been mentioned that circles in the Euclidean plane can not be defined by the focus-directrix property. If the conic is non-degeneratethen: Circles, not constructible by the earlier method, are also obtainable in this way. Conics may be defined over other fields that is, in other pappian geometries. The Euclidean plane may be embedded in the real projective plane and the conics may be considered as objects in this projective geometry.

The most ancient mathematical texts available are the clay tablet Plimpton c. Who is considered the first geometric theorems by logical reasoning such as: The Euclidean 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.

In particular two conics may possess none, two or four possibly coincident intersection points. This symbolic representation can be made concrete with a slight abuse of notation using the same notation to denote the object as well as the equation defining the object. These standard forms can be written parametrically as.

The semi-minor axis is the value b in the standard Cartesian equation of the ellipse or hyperbola. 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. After the time of Archimedes mathematics underwent a change influenced by the Romans who were interested only use mathematics to solve problems in their daily lives, who almost did not contribute to the development of this.

The linear eccentricity c is the distance between the center and the focus or one of the two foci. One of them is based on the converse of Pascal’s theorem, namely, if the points of intersection of opposite sides of a hexagon are collinear, then the six vertices lie on a conic.


The emergence of mathematics in human history is closely linked to the development of the concept of number, process happened very gradually in primitive human communities. It can also be shown [18]: It shall be assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice.

This is important for many applications, such as aerodynamics, where a smooth surface is required to ensure laminar flow and to prevent turbulence. Topics in algebraic curves. The association of lines of the pencils can be extended to obtain other points on the ellipse. 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.

The sought for conic is obtained by this construction since three points AD and P and two tangents the vertical lines at A and D uniquely determine the conic.

However, as the point of intersection is the apex of the cone, xecciones cone itself degenerates to a cylinderi. In the Euclidean plane, using the geometric definition, a degenerate case arises when the cutting plane passes through the apex of the cone. The type of the conic is determined by the type of cone, that is, by the angle formed at the vertex of the cone: 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.

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The focal parameter p is the distance from the focus or one of the two foci to the directrix. This equation may be written in matrix form, and some geometric properties can be studied as algebraic conditions. In this case, seccionfs plane will intersect both halves of the cone, producing two separate unbounded curves.

The conic sections have been studied by the ancient Greek mathematicians with this work culminating around BC, when Apollonius of Perga undertook a systematic study of their properties.