Area of a blob

[questionposter]

Newton devised some way to measure the area of a blob by using some infinite triangle theorem, but I have no idea what the theorem is right now.

[md65536]

Newton devised some way to measure the area of a blob by using some infinite triangle theorem, but I have no idea what the theorem is right now.

Either this is meant to be only moderately informative, or your name is curiously ironic.

 

It sounds kind of like integral calculus to me.

 

 

[Schrödinger's hat]

In my experience, surfaces are more frequently parameterised as some kind of quad when you're just after the area.

No reason triangles wouldn't work, though.

I seem to recall the derivation of a lot of calculus/geometric calculus formulae employing triangles.

[ewmon]

I'm guessing that this method is similar to the trapezoidal method of finding the area under a curve, but it forms triangles (like pie slices) instead of trapezoids. The method can find the area of a circle, and thus, the value of pi.

[questionposter]

I'm guessing that this method is similar to the trapezoidal method of finding the area under a curve, but it forms triangles (like pie slices) instead of trapezoids. The method can find the area of a circle, and thus, the value of pi.

 

I guess that's kind of what I was talking about, but I was thinking more about finding the area of a random blob using some kind of infinite triangle theorem.

[Schrödinger's hat]

I guess that's kind of what I was talking about, but I was thinking more about finding the area of a random blob using some kind of infinite triangle theorem.

 

To deal with any sort of limits or infinities you'd need to be able to describe your blob exactly as a function or series of functions.

Once you can do that, breaking it down into triangles is fine. Although I can't think of what the general scheme for parameterization would be off hand.

[ewmon]

I can't think of what the general scheme for parameterization would be off hand.

One method would involve the polar coordinate system and integrating within that system, as well as an algebraic equation that defines the shape of the blob's perimeter. Quite frankly, I've forgotten how to integrate in the polar coordinate system.

 

Another method would involve the polar coordinate system and integrating/summing within that system, although the shape of the blob would be graphically represented and measured.

1. As a guess, it would probably provide a more accurate answer by having the polar coordinate system originating near the blob's geometric center, which can be found using this simple method.
   
2. Next, draw radials emanating from the center of mass with equal angular proportions (for example, 20° or 10° ... or even 1° if the person feels ambitious). These "pie slices" of the blob will be converted into isosceles triangles, which require the two simple dimensions of height and base to find their areas.
   
3. The height can be the average of the distance along its two radials, or the distance from the center of mass down the triangle to the edge of the blob midway between the two radials.
   
4. The base can be measured directly from the diagram, or it can be calculated as a function of the angle formed by the two radials and the triangle's height (this last method seems more accurate, especially with very thin slices).
   
5. Having done this for all the "pie slices", each slice's area can be calculated.
   
6. Then all the areas for the pie slices can be summed together to form the area of the blob.
   

[*]If it's all the same to you, questionposter, I'd suggest you use a planimeter to find the area of blob. It's much more mysterious than Newton's infinite triangle method.

[DrRocket]

[*]If it's all the same to you, questionposter, I'd suggest you use a planimeter to find the area of blob. It's much more mysterious than Newton's infinite triangle method.

 

It is just a matter of integrating the appropriate differential form around the perimeter.

 

 

http://www.math.ntnu.no/~hanche/blog/planimeter.pdf