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In mathematical geometry, is a moving point the same as any other point?
Not the same thing.

Any point generally refers to a point that meets certain conditions and is within a certain range, focusing on the uncertainty of position.

A moving point generally refers to a point on an object that meets certain conditions and is concentrated in a variable position, that is, motion. Specifically, the moving point is generally on a line segment, a line or a ray. When the moving point moves, its sub-objects will also move, so the size and position of the object will also change.