Hi Micheal,

thanks so much for posting your question to this forum! I am happy to help you out any way I can, though I cannot, unfortunately, come out to Nelson as I am currently residing on the US East Coast. If I were still on Gabriola, well, then things might be different.
I am not surprised that you are finding yourself confused after reading the Nakahara book. It is intended as a general survey, just touching on various aspects of Japanese carpentry, and there are issues with the translation as it is. A good place for inspiration at least.
Since I don't know your grasp of hip roof framing generally, I am not sure how to advise you. The Japanese method of determining the cut angles for a regular hip are quite straightfoward in their own way, but it is nevertheless a complex topic. I f you could be more focused with your questions, I will help you as I can, but to explain the spectrum of Japanese hip roof framing in detail would be beyond this medium here I'm afraid.
The Japanese rafter square, sashigane, has a variety of scales on it, though some sashigane lack certain scales.
The front of the sashigane, omote-me (the 'me' is pronounced like 'may', in case you were wondering, and means 'scale'), has the conventional measure on it - whether in traditional sun and shaku, in inches, or in centimeters.
The back of the sashigane, ura-me, has several posssible scales. Typically it will have the kaku-me on the long arm, which is the front scale measure x sq.rt. 2 - this is the one used for regular hip rafter work primarily. This scale is the one that, if held across the diagonal end of a log (the top, not the butt) will tell you how large a square section log can be milled. There are transfer points from kaku-me to omote-me scales, and vice versa. Thus, if you look at 10 sun on the front scale, and note the mark at 10, you will see a mark carried over to the kaku-me, and it will read about 7.07 at that point (10 divided by the sq.rt. of 2). Conversely, if you took 10 on the kaku-me side, note the mark and turn it over to the omote-me side, you will see a mark at 14.14 - that is, 10 x sq.rt. 2. These carry over marks allow you to use the sashigane either side down most of the time.
The scale you mentioned, maru-me, relates diameter to circumference - it is a pi scale in other words. Hold this scale across the diameter of a circle, and the measurement you read will be the circumference of the circle. This scale has quite limited use and is sometimes omitted from certain squares. I decided to omit it from the square I designed a couple of years ago for Shinwa becasue it is just not particualrly useful. This scale is of no use for hip rafters, even if they are curved.
If you work in inches, I don't recommend the older inch-scale sashigane unless you plan to work in base 12. As Japanese roofwork is done in base 10, it is hand to have a sashigane that works on that basis. And if you are to work in base 10, it would be also handy to have a tape measure delineated in 10ths and 20ths too. I am thinking that Starrett might make one, but I'm not sure. If you intend to work in base 12, then a regular western framing square is the tool to use.
Okay, well, perhaps if you could post a drawing of the gate you are planning, or plan views, I could see more clearly what your ideas are.
I look forward to your next post Micheal.