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Geometría proyectiva: Obtaining conical shafts from two pairs Diameters Polar Conjugates

Los ejes de una cónica son aquellos diámetros polares conjugados que son ortogonales entre si.

We recall that two polar conjugate diameters, necessarily pass through the center O of the conical, are the polar two points unfit (located at infinity) that they are conjugated, namely, the polar of each of these points contains the other.

These pairs of elements determine an involution of diameters (polar) Conjugates will be defined when two pairs of beams know and their homologues.

Let us suppose that de una cónica se conocen, entre otros posibles elementos, dos parejas de diámetros y sus conjugados, for example a-a and b-b’.

El objetivo es encontrar la pareja de rectas homólogas que sean ortogonales entre sí. Para ello seccionaremos por una circunferencia que contenga al vértice del haz de rectas obteniendo una serie de segundo orden en involución que es proyectiva del haz de rectas. En esta serie de segundo orden podemos determinar el centro de involución I, ya que cada par de puntos homólogos en la involución estarán alineados con este punto.

Si quisiéramos obtener el elemento homólogo de cualquier punto de esta serie, su homólogo se encontrará sobre la circunferencia alineado con I. En particular si queremos encontrar dos rayos homólogos que sean ortogonales deberán cortar a la circunferencia en puntos de un diámetro (para la ortogonalidad) que contenga al centro de involución (para asegurar que son homólogos en la involución)

Esto nos permite obtener los ejes de la cónica en dirección, aunque faltará aún determinar la magnitud de los mismos.

Projective Geometry

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