'cause arbori are not boring

[BeholdTheTruth]

Happy New Year one and all (from an old neebie who has seen a lot more New Year days than most of you!)

And, if you do not mind me beginning near the end of my exploration of the nature of natural processes of flowings and ebbings, two math questions that have come up as I have more and more been playing around with some (what perhaps are not well known aspects of) arbelos-type unfoldings and complementary refoldings...

1) Consider two tangent circles (with, respectively, diameters AB and BC) inscribed within a circle having a diameter of AC. What is the formula for the portion of the area of the arbelos up to the point B, based on the value of diameter AB and the value of diameter BC?


2) Consider one circle AB tangent to and inscribed within another circle AC (I do not know the math term for such a duo.) What is the formula for the area of the arbelos-like portion (the larger circle's area less the smaller circle's area) at a given point within the transition from first to last instance of the tranformation as the difference between the areas of the inner and outer circles is gradually filled up beginning at the point of tangency and ending up at the point of tangency?

Looking forward to learning some really great answers. (I have heard so much about this forum from a very very close co-relation who thinks so highly of many of you.)

 

[Dr Adequate]

Everything you could possibly want to know about the arbelos.

In particular, they give this nice result for the area: draw the perpendicular BD from the tangent of the two circles to the edge of the large circle. Then the area of the arbelos is the same as the area of the circle with diameter BD.

I presume that it's this sort of constructive-geometric answer that you're looking for.

 

[casebro]

Strictly intuitively, 1/PI ?

 

[BeholdTheTruth]

[http: // mathworld.wolfram.com / Arbelos.html]Everything you could possibly want to know about the arbelos.

In particular, they give this nice result for the area: draw the perpendicular BD from the tangent of the two circles to the edge of the large circle. Then the area of the arbelos is the same as the area of the circle with diameter BD.

I presume that it's this sort of constructive-geometric answer that you're looking for.

Thanks for the pointer. BTW, do you happen to know of any correspondenses between the arbelos and the geometric aspects of the yin yang?