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Apparently for a 2d convex hull, the perimeter of it is pi times the mead width of it. Does that count as a circle being involved? I guess if you showed that

by drdeca 10y ago

Apparently for a 2d convex hull, the perimeter of it is pi times the mead width of it. Does that count as a circle being involved? I guess if you showed that the perimeter was proportional to the mean width, then circles would prove that the constant of proportion would be pi, because pi is (often? by definition) the perimeter of the circle divided by the (constant) width. I don't know if you count that as a circle being involved.

seandougall 10y ago
I would. Pi is (indeed always) the circumference of a circle divided by its diameter. To calculate the perimeter of a 2d convex hull, my impulse would be to model the shape in polar coordinates and integrate over theta. The fact that this calculation relies on polar coordinates strongly implies a relationship to circles -- specifically, L = 2 * pi * r_mean turns out to be unsurprising, since you're basically treating each slice of the hull as a slice of a circle, and calculation based on that circle's radius.
- drdeca 10y ago
  by usually I meant that is usually how it is defined. Of course it is always true that it is that value, I just meant, like, what description of the value is used to define what value we mean. I was unclear on that part. I meant that I think that in most cases, pi = circumference / diameter is the definition of pi, rather than pi being defined in some other way, and it being proven as a theorem that pi = circumference / diameter. Like, suppose I defined pi as sqrt(6*Zeta(2)) , this would still be the same value, so it would work as a definition of pi, but I don't think it is often defined in a way like this. I think I mostly wrote ^ because I felt like the way I worded my previous comment could be misinterpreted in a way that I feel would make me look dumb, so I wanted to clarify what I actually meant. Your point about considering it in polar coordinates seems like a good explanation. I hadn't thought of that, I like it, thank you for that. I don't yet see how to make that rigorous, but it sounds like it makes sense in a nice way. Thank you.

by drdeca 10y ago

seandougall 10y ago

I would. Pi is (indeed always) the circumference of a circle divided by its diameter. To calculate the perimeter of a 2d convex hull, my impulse would be to model the shape in polar coordinates and integrate over theta. The fact that this calculation relies on polar coordinates strongly implies a relationship to circles -- specifically, L = 2 * pi * r_mean turns out to be unsurprising, since you're basically treating each slice of the hull as a slice of a circle, and calculation based on that circle's radius.

drdeca 10y ago
by usually I meant that is usually how it is defined. Of course it is always true that it is that value, I just meant, like, what description of the value is used to define what value we mean. I was unclear on that part. I meant that I think that in most cases, pi = circumference / diameter is the definition of pi, rather than pi being defined in some other way, and it being proven as a theorem that pi = circumference / diameter. Like, suppose I defined pi as sqrt(6*Zeta(2)) , this would still be the same value, so it would work as a definition of pi, but I don't think it is often defined in a way like this. I think I mostly wrote ^ because I felt like the way I worded my previous comment could be misinterpreted in a way that I feel would make me look dumb, so I wanted to clarify what I actually meant. Your point about considering it in polar coordinates seems like a good explanation. I hadn't thought of that, I like it, thank you for that. I don't yet see how to make that rigorous, but it sounds like it makes sense in a nice way. Thank you.