The locations of the acoustic centers, and resulting path distance between them is based on research described by John Vanderkooy in several AES papers. At low frequencies (baffle and driver are small relative to the wavelength) the source can be treated as a point. However, polar measurements indicate that this point source is not located at the driver cone but rather a distance in front of the baffle/driver. The distance to the acoustic center is mainly defined by the size of the baffle, but is also affected by the size of the driver, and the depth of the cabinet. The space it is radiating into also has an effect. Move the woofer from free space down to the floor and then up against the wall (4-Pi to 2-Pi to 1-Pi) and the acoustic centers move further and further out in front of the woofer or H-frame/U-frame opening.I'm wondering how this smallest path is calculated? Because I would think the sound has to travel around the baffle which would lead to a larger number?

More details and comparison between measurement and theory posted here:
https://www.diyaudio.com/community/threads/hornresp.119854/page-500#post-5923396
If you are interested in using U-frame as a cardioid you might find these posts helpful:
https://www.diyaudio.com/community/threads/hornresp.119854/post-6166811
https://www.diyaudio.com/community/threads/hornresp.119854/post-6172980
If interested in the AES papers they can be found here:
The Low-Frequency Acoustic Center: Measurement, Theory, and Application (2010)
https://www.aes.org/e-lib/browse.cfm?elib=15289
Applications of the Acoustic Centre (2007)
https://www.aes.org/e-lib/browse.cfm?elib=14087
The Acoustic Center: A New Concept for Loudspeakers at Low Frequencies (2006)
https://www.aes.org/e-lib/browse.cfm?elib=13746
Polar Plots for Low Frequencies: The Acoustic Centre (2006)
https://www.aes.org/e-lib/browse.cfm?elib=13588