APOLONIO DE PERGA SECCIONES CONICAS PDF

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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Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines.

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In particular two conics may possess none, two or four possibly coincident intersection points. Traditionally it has been considered that mathematics as a science emerged in order to do calculations in commerce, to measure land and to predict astronomical events. In the corresponding affine space, one obtains an ellipse if the conic does not intersect the line at infinity, a parabola if the conic intersects the line at infinity in one double point corresponding to the axis, and a hyperbola if the conic intersects the line at infinity in two points corresponding to the asymptotes.

perag Elliptic curve cryptography Elliptic curve primality. However, as the point of intersection is the apex of the cone, the cone itself degenerates to a cylinderi. If the cutting plane is parallel to exactly one generating line of the cone, then the conic is seccciones and is called a parabola. Analytic theory Elliptic function Elliptic integral Fundamental pair of periods Modular form.

If there is only one intersection point, which has multiplicity 4, the two conic sections are said to be osculating. Conic sections are important in astronomy: The polar form of the equation of a conic is often used in dynamics ; for instance, determining the orbits of objects revolving about the Sun. There are some authors who define a conic as a two-dimensional nondegenerate quadric. An instrument for drawing conic sections was first described conicax CE by the Islamic mathematician Al-Kuhi.

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Conic section – Wikipedia

Teubner, Leipzig from Google Books: Non-degenerate conic sections are always ” smooth “. In the notation used here, A and B are polynomial coefficients, in contrast to some sources that denote the semimajor and ds axes as A and B. 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 conkcas equation in three variables or equivalently, the zeros of an irreducible quadratic form.

Apollonius used the names ellipseparabola and hyperbola for these curves, borrowing the terminology from earlier Pythagorean work on areas.

These are called degenerate conics and some authors do not consider them to be conics seccionfs all. An oval is a point set that has the following properties, which are held by conics: Counting points on elliptic curves Division polynomials Hasse’s theorem on elliptic curves Mazur’s torsion theorem Modular elliptic curve Modularity theorem Mordell—Weil theorem Nagell—Lutz theorem Supersingular elliptic curve Schoof’s algorithm Schoof—Elkies—Atkin algorithm.

There are three types of conics, the ellipseparabolaand hyperbola. seccioes

Various parameters are associated with a conic section. In the case of the parabola, the second focus needs to be thought of as infinitely far away, so that the light rays going toward or coming from the second focus are parallel.

For specific applications of each type of conic section, see CircleEllipseParabolaand Hyperbola. 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 three types are then determined by how this line at infinity intersects the conic in the projective space. In polar coordinatesa conic section with one focus at the origin and, if any, the other at a negative value for an ellipse or a positive value for a hyperbola on the x -axis, is given by the equation.

Treatise on conic sections

Although possessed some ability to estimate sizes and magnitudes, they apolonuo initially had a notion of number. It scciones also be shown [18]: A circle is a limiting case and is not defined by a focus and directrix, in the plane however, see the section on the extension to projective planes. Aoolonio a projective space over any division ring, but in particular over either the real or complex numbers, all non-degenerate conics are equivalent, and thus in projective geometry one simply speaks of “a conic” without specifying a type.

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Furthermore, each straight line intersects each conic section twice. Thus there is a 2-way classification: 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.

Sección cónica

In the real projective plane, since parallel lines meet at a point on the line at infinity, the parallel line case of the Euclidean plane can be viewed as intersecting lines. In the case of an ellipse the squares of the two semi-axes are given by the denominators in the canonical form.

The greatest progress pergz the study of conics by the ancient Greeks is due to Apollonius of Perga died c.

Let C 1 and C 2 be two distinct conics in a projective plane defined over an algebraically closed field K. With this terminology there are no degenerate conics only degenerate quadricsbut we shall use the more traditional terminology and avoid that definition. After Thales, Pythagoras gives an advance mathematics with discoveries like the Pythagorean theorem, the discovery of irrational; also “The Pythagoreans developed a first group of four mathematical disciplines: Then Archimedes and Apollonius appears.

Karl Georg Apolonnio von Staudt defined a conic as the point set given by all the absolute points of a polarity that has absolute points.