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The book Karate Science includes the statement that:

One of the main goals of karate is to have the body postured in such a way as to allow a direct connection from the floor to the striking limb at the point of impact.

This is accompanied by the following diagram (with numbers rather clumsily added by me for clarity in the question):

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However, I don't understand how any of these represent a "direct connection." Line 2 goes to the center of the body and only connects to the ground indirectly via 3, 4, and 5.

I also don't see how line 1 exists at all (except as an "imaginary line" brought about by the alignment of the arm and front leg).

Can someone explain this diagram to me? In particular, where the "direct connection" with the floor?

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From a non-karate perspective: the direct connection is better defined by force transfer than stance geometry. Imagine someone pushing back into your punching arm. If you maintain your shape and balance as this force is applied, either the force will get transferred instantaneously into the ground or you will slide. When you remain in place (don't slide), this is a direct connection, at least in my mind. Contrast this with what happens when a joint collapses (whether arm, leg, back, etc.); now a segment of your body is moving and the force transfer is not direct.

I can't speak to what is happening in the diagram.

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With the caveat that I only have a year or two experience in Karate (albeit four years in Tang Soo Do), I was taught that the key lines are 2 and 4, ideally forming a straight line from the bracing/pushing back leg to the fist. Line 3 is showing the spine, which connects the two since we humans don't have arms connected to our legs directly. Line 5 is what keeps you from falling as you throw your body weight forward. Line 1 shows that your force is over that front knee so that, ideally, you are never off-balance, and that balancing leg isn't holding back any of the force.

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