Distance between lines in cheat sheet

regarding the formula in https://bivector.net/3DPGA.pdf I have a question about distance between lines

The sheet mentions :

d_{l1,l2} = \csc \alpha ( \hat{l_1} \vee \hat{l_2} )

I’m a bit triggered by this \cosec \alpha here… I thought that the join of two lines already measured (roughly speaking) the signed distances between the lines measured along their common normal line.

Here it is as if the weight of the meet was proportional to the oriented perpendicular distance between the two lines but weighted by the sine of the angle between their relative directions.

Btw this formula will not work for distance between parallel lines, shouldn’t there be another one for this case implying something like the infinite norm of their wedge product ?

Yes, the meet is weighted by the sine, it really computes difference in location and angle combined. The lines are only the same if both aspects are zero. And in the degenerate case, the join is not the full space, so the meet is redefined relative to what is now the join. All a bit awkward, the meet is a ‘piecewise linear’ form of intersection.

Thanks for this clarification !