# How to visualize the pole/polar of a line/point in projective space PG(2, R) not in contact with the curvature?

So, I understand that given a curvature $$\mathbb{C}$$. The polar of a point on the conic is tangent to the conic at that point and the pole of a line touching the conic only once and tangent to the conic is the point where it touches.

I'm failing to understand what the polar of internal and external points are and the poles of secant lines and external lines are. It is stated that an external point has a secant polar and an internal point has an external line as a polar.

Could someone explain why this is? How does this work intuitively?

• Is your “curvature” a conic? – amd Oct 17 '18 at 20:36
• Yes, it is a conic. – Neo Oct 18 '18 at 5:29 In this figure, $$A$$ is the pole of $$a$$, $$B$$ of $$b$$ and $$C$$ of $$c$$. Both $$(B,b)$$ and $$(C,c)$$ have the point outside and the polar line a secant and thus can be paired via the tangents (dark green) and points of contact (light green). The pair $$(A,a)$$ then follows from that due to incidence conservation.