This crossed my mind today.
I am not Veeky Forums person so I hope that you already know the answer.
How do we calculate surface area of pic related?
Second example would be surface area of countries or any other irregular curved shapes.
If you could post some basic example it would be great.
bump for knowledge
numerical integration is pretty handy for irregular shapes, grab a pen and a piece of paper and get started OP
But seriously how do they calculate areas of countries? :(
put it on a grid and count a number of sqaure units that lie within this shape
the finer the grid is, the more accurate result you get
you can also plot a function S=f(unit size) and eyeball the limit of this function, or maybe obtain an approximated analytic representation of the function which would converge where unit size approaches zero (1 divided by positive infinity)
TLDR: brute force
Seriously that's how they do it, except they use a computer obviously
Thanks. Its awesome that we still cant calculate stuff to their actual real world sizes , isntead we approximate a lot dont we and it still works
The coastline of Britain is infinite
isn't coastline different from area a la Gabriel's Horn?
You're right. The area would converge. I am reminded of Kochs curve.
In your math fetish world, it would be, but distance resolution is ultimately finite in the real world
Run a string around the perimeter lol
Depends. Can we always find a smaller resolution? Can you prove we've discovered the smallest resolution?
We cannot talk about meaningful measurements beyond the Planck length, so yes.
when you reach distances that small, its more interesting to start defining what counts as the coastline
Would the equation (perimeter/4)^2 work?
The coastline problem is cool as fuck.
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>cannot talk about
Cannot disprove their existence?
Wiki "Planimeter". If a mechanical device can do it then it can be simulated with a computer program. Then just feed in a list of coordinates for the perimeter and get the area. The more the points, the better the accuracy.
Reading this thread and some comments reminded me of Hausdorff dimension. In particular, coastlines are more fractal in nature (as opposed to being like a line or like
'an area'). For example they numerically calculated the coastline of Ireland to be of Hausdorff dimension approx 1.22.
en.wikipedia.org
>perimeter of a circle = 2*pi*r
>perimeter/4= 1/2*pi*r
>(perimeter/4)^2= 1/4*pi^2*r^2
1/4*pi^2*r^2 != pi*r^2
No. Your formula only works for squares.
Cannot assert their existence either.
The approximations can be more accurate than if you had a dude with a tape measure walking around
Count the number of white pixels inside the line.