Posting this here, from an FKC thread that @Daun_yon started, cause I think its interesting, and it will soon be lost in the facebook abyss. I just screen grabbed the replies that were attempts at answering the question:
Further thoughts: Hmm actually, since the final object isn't a perfect sphere (the bevel is cut out as a 'flat spot' along the circumference), should the equation be: Ratio: (SA of circular bevel) / (SA of sphere - SA of cap + SA of circular bevel) Rather than: (SA of cap) / (SA of sphere) ?
So I guess if your going to increase your bevel size you should increase your cup sizes in accordance? To stop your lighthouses tripping over in the bevel. I wonder if the JKA worked it like this when originally designing their current competition kendamas?
@Stuart Barron yea, that makes sense. Larger tama bevel with the same size bottom cup means higher chance of lighthouse tipping over. Could probably work to determine the ratio there as well, to find the point at which the tipping becomes an issue. It would be interesting to know the conversations that the JKA board had to design the original JKA shape, thats for sure!
On the base cup and tipping issue, from my experience (no math involved tho), it's not actually as big of an issue as you would think. The LunaTac XL Trainer Tamas have a Massive bevel, like biggest you've ever seen. It's just slightly smaller than the base cup on most kendamas. Even with a bevel that large, I rarely notice it affecting lighthouses. The only time it does affect it is if you land one at an angle. Other than that, the lighthouse tends to just slide right over it without tipping in.