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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This equation may be written in matrix form, and some geometric properties can be studied as algebraic conditions. On the other hand, starting with the real projective plane, a Euclidean plane is obtained by distinguishing some line as the line at infinity and removing it and all its points.

The midpoint of this line segment is called the apolknio of the conic.

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If the conic is non-degeneratethen: Von Staudt introduced this definition in Geometrie der Lage as part of his attempt to remove all metrical concepts from projective geometry. In standard form the parabola will always pass through the origin.

If the intersection point is double, the line is said to be tangent and it is called the tangent line. The most general equation is of the form [12].

Sección cónica

Johannes Kepler extended the theory of conics through the ” principle of continuity “, a precursor to the concept of seccioned. 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. The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone — for a right cone, see diagram, this means that the cutting plane is perpendicular to the symmetry axis of the cone.

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.

It can also be shown [18]: An instrument for drawing conic sections was df described in CE by the Islamic mathematician Al-Kuhi. The circle is a special case of the ellipse, and is of sufficient interest in its own right that it was sometimes called a fourth type of conic section.


If they are bound together, they will both trace out ellipses; if they are moving apart, they will both follow parabolas or hyperbolas. De Franchis theorem Faltings’s theorem Hurwitz’s automorphisms theorem Hurwitz surface Hyperelliptic curve. 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 quadratic equation in three variables or equivalently, the zeros of an irreducible quadratic form. Conic sections are important in astronomy: If the points at infinity are 1,i,0 and 1,-i,0the conic section is a circle see circular points at infinity.

These are called degenerate conics and some authors do not consider them to be conixas at all. Various parameters are associated with a conic section. Science in medieval Islam: 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 the right triangle.

Conic section – Wikipedia

It has been mentioned that circles in the Euclidean plane can not be defined by the focus-directrix property. From Wikipedia, the free encyclopedia. I, Dover,pg. It can be proven that in the complex projective plane CP 2 two conic sections have four points in common if one accounts for multiplicityso there are never more than 4 intersection points and there is always one intersection point possibilities: 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.

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

This may account for why Apollonius considered circles a fourth type of conic section, a distinction that is no longer made. Cones were constructed by rotating a right triangle about one of its legs so the hypotenuse generates the surface of the cone such a line is called a generatrix.

Four points in the plane in general linear position determine a unique conic passing through the first three points and having the fourth point aolonio its center. In this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. The three types are then determined by how this line at infinity intersects the conic in the projective space. The most cobicas mathematical texts available are the clay tablet Plimpton c. Greek mathematics contributed in the geometric language, all knowledge of elementary mathematics, that is, on the one hand the synthetic plane geometry points, lines, polygons and circles and spatial planes, polyhedra and round bodies ; and on the other hand, an arithmetic and algebra, both with a geometric clothing, contributions that were made in the book concas Elements” of Euclid.


In analytic geometrya conic may be defined as a plane algebraic curve of degree 2; that is, as the set of points whose coordinates satisfy a quadratic equation in two variables. 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 the complex projective plane the non-degenerate conics can not be distinguished from one another. In the real projective plane, a point conic has the property that every line meets it in two points which may coincide, or may be complex and any set of points with this property is a point conic.

Treatise on conic sections

Wikimedia Commons has media related to Conic sections. Views Read View source Escciones history. 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.

The emergence of mathematics in human history is closely linked to the concias of the concept of number, process happened very gradually in primitive human communities.

This can be done for arbitrary projective planesbut to obtain the real projective plane as the extended Euclidean plane, some specific choices have to be made. The labeling associates the lines of the pencil through A with the lines of the pencil through D projectively but not perspectively. Thus, the numbers beyond two or three, had no name, so they used some expression equivalent to “many” to refer to a set older. That is, there is a projective transformation that will map any non-degenerate conic to any qpolonio non-degenerate conic.

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 conifas.