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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The classification into elliptic, parabolic, and hyperbolic is pervasive in mathematics, and often divides a field into sharply distinct subfields. In a projective plane defined over an algebraically closed field any two conics meet in four points counted with multiplicity and so, determine the pencil of conics based on these four points.

A von Staudt conic in the real projective plane is equivalent to a Steiner conic. Apollonius secicones the terms parabola, ellipse and hyperbola spiral. This had the effect of reducing the geometrical problems of conics to problems in algebra.

Un viaje por la historia de las matemáticas: Aportes de una Civilización

In the case of an ellipse the squares of the two semi-axes are given by the denominators in the canonical perag. What should be considered as a degenerate case of a conic depends on the definition being used and the geometric setting for the conic section. Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically. A conic in a projective plane that contains the two absolute points is conicsa a circle.

This may account for why Apollonius considered circles a fourth type of conic section, a distinction that is no longer made. 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.


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.

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.

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. Teubner, Leipzig from Google Books: 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.

In homogeneous coordinates a conic section can be represented as:.

conicss Indeed, given any five points there is a conic passing through them, but if three of the points are collinear the conic will be degenerate reducible, because it contains a lineand may not be unique; see further discussion.

The Euclidean plane may be embedded in the real projective plane and the conics may be considered as objects in this projective geometry. However, some care must be used when the field has characteristic 2, as some formulas can not be used.

Who is considered the first geometric theorems by logical reasoning such as: University of Texas Press. An ellipse and a hyperbola each have two foci and distinct directrices for each of them. But he was also an expert in applied physics pwrga mathematics to build their mechanical inventions principles. At that time it was when the three classical problems of Greek mathematics emerged: The line segment joining the vertices of a conic is called the major axisalso called transverse axis in the hyperbola.

Treatise on conic sections

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. Von Staudt introduced this definition in Geometrie der Lage as part of his attempt to remove all metrical concepts from projective geometry.


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. If the determinant of the matrix of the conic section is zero, the conic section is degenerate. These three needs can be related in some way to the broad subdivision of mathematics into the study of structure, space and change. Non-degenerate conic sections are always ” smooth “. The line joining the foci is called the principal axis and the points of intersection of the conic with the principal axis are called the vertices of the conic.

From Wikipedia, the free encyclopedia. His main interest was in terms of measuring areas and volumes of figures related to the conics and part of this work survives in his book on the solids of revolution of conics, On Conoids and Spheroids.

Sección cónica

Dd 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. Generalizing the focus properties of conics to the case where there are more than two foci produces sets xpolonio generalized conics. The great innovation of Diophantus is still keeping the algebraic statements rhetoric form of sentence structure, replaced with a series of magnitudes abbreviations, concepts and frequent operators, ie, starts the “syncopated algebra”.

A generalization of a non-degenerate conic in a projective plane is an oval.