How to visualize the pole-polar relationship if placing pole inside the conics?

Question

This problem comes from R.Hartley & A.Zisserman Multiple View Geometry in Computer Vision at page 58:

To my understanding, the polar $l$ dipicted in above figure goes through two points at which tangent lines insect at pole x. The polar is tangent line when x is on the conic.

My question: How to visualize polar $l$ when x is inside the conic?

Answer 1

The pole-polar has the following important property: if point $x$ is on line $l$ then the polar of $x$ contains the pole of $l$.

So, to find the polar of a point inside, take two line passing through $x$, $l_1$ and $l_2$. Find the poles of those lines, $x_1$, and $x_2$. The line through $x_1$, $x_2$ is the polar of $x$.

Answer 2

You simply do not visualize any such thing. There are no tangents to the conic from a point inside.

Answer 3

Here is something I hope that can be useful