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Geometría proyectiva: Obtaining the conical center

By studying the concepts of Conjugate directions We saw a definition for center-based conical basic concepts of polarity:

The center of the cone is the pole of the improper line.

For the conical center will need to have poles and polar respect thereof. In particular constructions are simplified if we know tangents and contact points. We will see that is especially immediately if three tangents and their respective contact points are known, obtained from the definition of the conic by 5 data and application of the techniques disclosed to determine tangents and points of tangency:

Therefore we consider that are three tangents and their respective contact points, determined from the above procedures.

Polarity

If we consider the involution between overlapping series of second order with homologous pairs A-A’ and B-B’, point And es el Center of involution and the line and the axis of involution. The line “and” is polar point “And” with respect to the straight lines “r” and “s”.

Points “T1” and “T2” doubles in this involution and therefore, the tangents to the conical them pass through the center “And” involution. Therefore:

The polar “and” to a point “And” through the points of tangency “T1” and “T2” the tangents to the tapered from “And“, since it is the shaft center involution And.

In this figure we can see that the polar point “E1” It is straight “e1“. Bastaría suponer que E1 es el centro de una involución que transforma el punto A on B and point A‘ on B‘, thus es armónica la cuaterna (E1 E2 T1 T2)

A partir de esta cuaterna armónica podemos concluir propiedades interesantes para determinar el centro de la cónica. It E1 es un punto impropio el punto E2 deberá de ser el punto medio entre T1 and T2 . En consecuencia la recta E-E2, polar de E1, deberá contener al centro de la cónica

En el caso en que el punto “And” sea impropio (at infinity), las tangentes desde este punto serán paralelas y la recta “and” se convertirá en un diámetro de la cónica pasando por el centro de la misma.

Lugar geométrico del centro de la cónica

La obtención del centro se realizará mediante la intersección de dos lugares geométricos obtenidos a partir del mismo principio. Analizaremos este lugar geométrico para el que necesitamos dos tangentes y sus puntos de tangencia.

Para determinar el lugar geométrico buscado buscaremos el punto medio entre dos puntos de tangencia, ya que esta recta es la polar del punto I de intersección de las tangentes en dichos puntos. Como hemos visto, la recta que pasa por este punto medio y el de intersección de las tangentes contiene al centro de la cónica.

El centro se obtendrá como intersección de dos lugares geométricos, repitiendo el proceso anterior para otra pareja de puntos de tangencia.

Projective Geometry

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