I just started to study about manifolds and there is something I miss about how the use of this math in a real scenario. I think the following problem will clarify my doubts and I hope your answer won't be too difficult. Imagine you are an ant (that lives in 2 dimensions) on a very big sphere and for some reason you want to describe the sphere with 2D cartesian coordinates. For every point there is a tangent plane and for this reason you can map every infinitesimal region of the sphere on some plane. I'm ok with the concept: it's like how humans use many geographic maps to describe the earth.
What I don't understand is how the ant can translate everything in formulas. For example if the ant want to describe a curve along the sphere how would it do? Obviously in this example the only things that the ant can use are the instrument to measure points on a 2D cartesian coordinate system. The ant can't use the equation of the sphere in 3D space.