Local tangent plane coordinates

Imagine you are standing on a big ball, like a giant beach ball. If you try to walk in a straight line, you will eventually end up going in circles because the ball is round.

Now imagine you have a flat piece of paper that you lay on that ball. If you place the paper just right, it will touch the ball at one point and be flat at that point. This is called the tangent point. The paper is like a small piece of the ball, but it's flat, so you can walk on it easily.

Now suppose you want to measure where things are on the flat paper. You could use a ruler or a grid, just like you would use for a flat piece of paper on a table. These measurements are called local tangent plane coordinates.

It's like you're looking at a tiny piece of the beach ball through a microscope and taking measurements of it using a ruler. The measurements are just for that small piece, not for the whole beach ball. That's what local tangent plane coordinates are all about - measuring a small, flat piece of a curve or surface.