"How to calculate the flat part of surface on the sphere" explanations below.

[Anton_Bazhenov]

If you are a master of math, pls help me to find information about "How to calulate the flat part of surface on the sphere". Some people say that flatness can't exist on "perfect" round sphere in theory but practicaly it is possible. For dumb example just to understand we have football fields on Earth, they are flat. The question is How much is a flat part possible on Eath or how to calculate this part of surface of sphere, at least on perfect round?

 

[blamocur]

If you are a master of math, pls help me to find information about "How to calulate the flat part of surface on the sphere". Some people say that flatness can't exist on "perfect" round sphere in theory but practicaly it is possible. For dumb example just to understand we have football fields on Earth, they are flat. The question is How much is a flat part possible on Eath or how to calculate this part of surface of sphere, at least on perfect round?

If you want your football field on a perfect sphere to be perfectly flat you will have to shovel some soil. Fortunately, for the case of the American football field (length 110m) the deepest you will have to dig is less than 0.25mm

 

[Dr.Peterson]

If you are a master of math, pls help me to find information about "How to calulate the flat part of surface on the sphere". Some people say that flatness can't exist on "perfect" round sphere in theory but practicaly it is possible. For dumb example just to understand we have football fields on Earth, they are flat. The question is How much is a flat part possible on Eath or how to calculate this part of surface of sphere, at least on perfect round?

That depends entirely on how flat you require something to be in order to be "practically flat". You get to decide on the required precision.

On a perfect sphere, no region can be exactly planar; a plane will touch it at a single point. So there is no actually "flat part of the surface of a sphere".

But if you specify how far you allow points on the sphere to deviate from a plane (as @blamocur has suggested), you could work out the allowed distance.

On the other hand, it may be worth considering that what we call "flat land" (a plain) might be just at a constant distance from the center (altitude), and therefore would be part of a sphere, not a plane.

 

[Anton_Bazhenov]

That depends entirely on how flat you require something to be in order to be "practically flat". You get to decide on the required precision.

On a perfect sphere, no region can be exactly planar; a plane will touch it at a single point. So there is no actually "flat part of the surface of a sphere".

But if you specify how far you allow points on the sphere to deviate from a plane (as @blamocur has suggested), you could work out the allowed distance.

On the other hand, it may be worth considering that what we call "flat land" (a plain) might be just at a constant distance from the center (altitude), and therefore would be part of a sphere, not a plane.

Thank you for your response. As I see the answer is zero. Now I understand the topic deeper. So practically the mathematical error of the arc can be very insignificant, down to the atoms, that we can consider it as a flat surface.
I'll try to explain why I asked this question. What if we consider the practical situation where two perfect spheres touch each other in one point, but they don't damage each other not to create some planar, so only two atoms of each sphere can touch each other provided that one atom is counted as undivided particle represents as zero segment. If we don't consider it as zero segment, which practically really strange doing it, so we can make a conclusion that two spheres can't touch each other without some damages, even on atom level. It makes me thinking how really math works. There are so many factors I can forget to account in calculations or collected data isn't complete. So my thoughts is not about math only, it's physics and astrophysics then.

 

[Dr.Peterson]

Thank you for your response. As I see the answer is zero. Now I understand the topic deeper. So practically the mathematical error of the arc can be very insignificant, down to the atoms, that we can consider it as a flat surface.
I'll try to explain why I asked this question. What if we consider the practical situation where two perfect spheres touch each other in one point, but they don't damage each other not to create some planar, so only two atoms of each sphere can touch each other provided that one atom is counted as undivided particle represents as zero segment. If we don't consider it as zero segment, which practically really strange doing it, so we can make a conclusion that two spheres can't touch each other without some damages, even on atom level. It makes me thinking how really math works. There are so many factors I can forget to account in calculations or collected data isn't complete. So my thoughts is not about math only, it's physics and astrophysics then.

The answer is very simple: You can't make a perfect sphere out of atoms!

Mathematics deals with idealized versions of the real world, and you can't equate the two.