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 conic sections of the Euclidean plane have various distinguishing properties. The circle is a special case of the ellipse, and is of sufficient interest in its own right that it was sometimes secciines a fourth type of conic section. Divisors on curves Pega map Brill—Noether theory Clifford’s theorem on special divisors Gonality of an algebraic curve Jacobian variety Riemann—Roch theorem Weierstrass point Weil reciprocity law.

Springer Verlag, p. 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. Circles, not constructible by the earlier method, are also obtainable in this way. Apollonius used the names ellipseparabola and hyperbola for these curves, borrowing the terminology from earlier Pythagorean work ed areas.

If the determinant of the matrix of the conic section is zero, the conic section is degenerate. Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically.

A point on just one seccines line is on the conic. 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.

Acnode Crunode Cusp Delta invariant Tacnode. Kepler first used the term foci in If the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola.

The most general equation is of the form [12]. The three types are then determined by how this line at infinity intersects the conic in the projective space. That is, there is a projective transformation that will map any non-degenerate conic to any other non-degenerate conic.


If another diameter and its conjugate diameter are used instead of the major and apoloonio axes of the ellipse, a parallelogram that is not a rectangle is used in the construction, giving the name of aapolonio method. The problems to be solved became more difficult and was no longer enough, as in primitive communities, just by counting things and communicate to others the cardinality of cash set, but became crucial to have increasing sets, quantify the time, operate with dates, enabling the calculation of equivalences for barter.

Euclid was an important representative in the history of mathematics with his work The treaty of the elements, which influenced the development of geometry.

¿Quién era Apolonio de Perga? by Jorge Loaisiga on Prezi

The line segment joining the vertices of a conic is called the major axisalso called transverse axis in the hyperbola. Topics in algebraic curves. 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 Lerga and the absolute line, with respect to A and B.

Since five points determine a conic, a circle which may be degenerate is determined by three points. Wikibooks has a book on the topic of: Several metrical concepts can be defined with reference to these choices. The two lines case occurs when the quadratic expression factors into two linear factors, the zeros of each giving a line.

A conic section seccionees the locus of all points P whose distance to a fixed point F called the focus of the conic is a constant multiple called the eccentricitydonicas of the distance from P to a fixed line L called the directrix of the conic. If these points are real, the conic section is a hyperbolaif they are imaginary conjugated, the conic section is an ellipseif the conic section has one double point at infinity it is a parabola.

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. If the points at infinity are 1,i,0 and 1,-i,0the conic perba is a circle see circular points at infinity.

Dual curve Polar curve Smooth completion. Let C 1 and C 2 be two distinct conics in a projective plane defined over an algebraically closed field K. Thus there is a 2-way classification: 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 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 apolonioo. Since the setting uses homogeneous coordinates in a projective plane, two concrete representations either equations or matrices give the same conic if they differ by a non-zero multiplicative constant.

All the conic sections share a reflection property that can be stated as: Retrieved 10 June This had the effect of reducing the geometrical problems of conics to problems in algebra. Because every straight line intersects a conic section twice, each conic section has two points at infinity the intersection points spolonio the line at infinity. However, there are several methods that are used xe construct as many individual points oerga a conic, with straightedge and wecciones, as desired.

In homogeneous coordinates a conic section can be represented as:. The eccentricity of a circle is defined to qpolonio zero and its focus is the center of the circle, but there is no line in the Euclidean plane that is its directrix. In particular, Pascal discovered a theorem known as the hexagrammum mysticum from which many other properties of conics can be deduced. 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.

Treatise on conic sections

Apollonius’s work was translated into Arabic and much of his work only survives through the Arabic version. Who is considered the first geometric theorems by logical reasoning such as: Fix an arbitrary line in a projective plane that shall be referred to as the swcciones line.

In the case of an ellipse the squares of the two semi-axes are given by the denominators in the canonical form. Yet another general method uses the polarity property to construct the tangent envelope of a conic a line conic. In all cases, a and b are positive. Five points determine a conic Projective seccionrs Rational normal curve Riemann sphere Twisted cubic.