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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If the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola.

Conic section – Wikipedia

If the conic is non-degeneratethen: This equation may be written in matrix form, and some geometric properties can be studied as algebraic conditions. Von Staudt introduced this definition in Geometrie der Lage as part of his attempt to remove all metrical concepts from projective geometry.

Views Read View source View history. Thus there is a 2-way classification: Mathematics and its history 3rd ed. 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.

In 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.

It secciones considered that Greek mathematics begin with Thales of MiletusB. If C 1 and C 2 have such concrete realizations then every member of the above pencil will as well.

Sección cónica

A pencil of conics can represented algebraically in the following way. From Wikipedia, the free encyclopedia. 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.

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

These parameters are related as shown in the following table, where the standard position is assumed.

A proof that the conic sections given by the focus-directrix property are the same as those given by planes intersecting a cone is facilitated by the use of Dandelin spheres. After conicxs 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 conica of this.


Divisors on curves Abel—Jacobi 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. 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 conic section is 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 eccentricitye of the distance from P to a fixed line L called the directrix of the conic. I, Dover,pg. In particular, Pascal discovered a theorem known as the hexagrammum mysticum from which many other properties of conics can be deduced. This had the effect of reducing the geometrical problems of conics to problems in algebra.

Apollonius’s study of the properties seccioned these curves made it possible to show that any plane cutting a fixed double cone two nappedregardless of its angle, will produce a conic according to the earlier definition, leading to the definition commonly used today.

The association conicaw lines of the pencils can be extended to obtain other points on the ellipse. However, some care must be used when the field has characteristic 2, as some formulas can not be used. Three types of cones were determined by their vertex angles measured by twice the angle formed by the hypotenuse and the leg being rotated about in scciones right triangle.

The conic sections have been studied for thousands of years and have provided a rich source of interesting and beautiful results in Euclidean geometry. If they are bound together, they will both trace out ellipses; if they are moving apart, they will both follow parabolas or hyperbolas.

There are some authors who define a conic as a two-dimensional nondegenerate quadric. A conic can not be constructed as a continuous curve or two with straightedge and compass.

Sección cónica – Wikipedia, a enciclopedia libre

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. Because every straight line intersects a conic section twice, each conic section has two points at infinity the intersection points with the line spolonio infinity.

A generalization of a non-degenerate conic in a projective plane is an oval. The classification into elliptic, parabolic, and hyperbolic is pervasive in mathematics, and often divides a field into sharply distinct subfields. A point on no tangent line is said to be an interior point or inner point of the conic, while a point on two tangent lines is an exterior point or outer point.


The conic section was then determined by apoolnio one of these cones with a plane drawn perpendicular to a generatrix. For example, the matrix representations used above require division by 2. Retrieved 10 June If another diameter and its conjugate diameter are used instead of the major and minor axes of the ellipse, a parallelogram that is not a rectangle is used in the construction, giving the name of the method.

It shall be assumed that the cone is a right circular cone for the purpose secicones easy description, but this is not required; any double cone with some circular cross-section will suffice. However, as the point of intersection is the apex of the cone, the cone itself degenerates to a cylinderi. The linear eccentricity c is the distance between the center and the focus or one of the two foci. The above equation can be written in matrix notation as [13].

At every point of a point conic there is a unique tangent line, and eecciones, on every apoolonio of a line conic there is a secciines point called a point of contact.

The circle is a special kind of ellipse, although historically it had been considered as a fourth type as it was by Apollonius. Deighton, Bell, and Co. It can be proven that in the complex projective plane CP 2 two conic sections have four points in sexciones if one accounts for multiplicityso there are never more than 4 intersection points and there is always one cinicas point possibilities: The eccentricity of an ellipse can be seen as a measure of how far the ellipse deviates from being circular.

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. In the complex projective plane the non-degenerate conics can not be distinguished from one another. At that time it was when the three classical problems of Greek mathematics emerged: In standard form the apooonio will always pass through the origin. The theorem also holds for degenerate conics consisting of two lines, but paolonio that case it is known as Pappus’s theorem.