$\displaystyle \sideset{}{_\theta}\prod$ is the orthogonal projection onto the line Lθ passing through the origin that makes an angle θ with the horizontal.

It's description is

"Count the number of intersections of a given set F with a straight line, and then integrate this number over the space of all lines."

But I think $\displaystyle \sideset{}{_\theta^{-1}}\prod$(y) is not a straight line and $\displaystyle \sideset{}{_\theta^{-1}}\prod$ is a set containing set F.

Also, I see there is only one staight line Lθ here.

What 's wrong in my thought?

(additional question: When F is a rectifiable curve, I1(F) is twice its length.Why?)