Theoretically it is said that, tangent touches to a single point on a circle. But If my circle is very big, and large enough, then i think, it should not be a just single point where my tangent is touching, though is will be a very small portion depending on how large is the circle. If i have a perfect sphere of size of earth, then a perfectly flat surface of size of football field will completely be touching on to the earth's surface, and is not just at one point! So, my question is, how big should be the radius of a circle, to perfectly allow 1 meter of area of a tangent touching perfectly on it?
If you have a perfect circle, then a tangent line will only touch the circle at one single point. It doesn't matter how large the circle is.
A circle is a circle. Small or large does not matter because both the border of the circle and the tangent line are zero width.
Maybe another way to look at this is: how close do you need to be to the surface of the circle/sphere for it to appear flat?
at a small enough scale, a circle is identical to a line; i am referring to a differential scale... however, a tangent will never intersect any curve more than once.
A circle is never identical to a line. Or do you mean an infinitesimal scale? This is false, but it is true for a circle.
that is what I meant by a differential scale also, you are right, a tangent can intersect a cubic curve in more than one point, and that is just one case...
from a topological point of view, both structures are infinite; without beginning or end. ie the "endpoints" of a line can coincide at infinity, and thus form a closed loop topologically equivalent to a circle. Furthermore, any segment of the line WILL contain the "center" of the line. I therefore propose that this center is the intersection of the endpoints of the line
that is why I used quotes, they are not really points that terminate the line. they are more like the boundary of infinity; two coincident lines can "grow" at different rates, and the line that grows fastest will enclose the other line. The enclosed line would have endpoints within the outer line, as it is entirely contained in the outer line.