CO-Tf2 Conical Pole

The Conical Pole P of a line L wrt some conic CO is the intersection point of the tangents to CO at the intersection points of CO with L. In this construction P is called the pole and L is called the polar.

This construction is very intuitive. However there is a flaw in the definition because the pole cannot be constructed under all circumstances. For example when CO is some ellipse and L is not intersecting the ellipse, then the construction fails.

Therefore another construction is made that includes the result of the first construction:

The pole of some line L wrt some conic (see CO-Tf1) is the intersection point of the polars of two random points from line L.

Another construction

This construction is from Eckart Schmidt. See Ref-34, QFG#2811.

1. Let L be the line to be transformed.
2. Let Pl be the pole (Co-Tf2) of L.
3. Let Lp be the line through Pl perpendicular to L.
4. Let Lc be the reflection of P.Co-P1 in the main axis of Co.
5. Then Co-Tf4(L) is the polar of the intersection point of Lp and Lc.

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