The curvature of an arc is the same everywhere, that is to say, it is a constant, so it is meaningful to define it, which is curvature. The curvature of an arc is defined as the radian per arc length, which is numerically equal to the reciprocal of the arc radius. The smaller the radius, the more curved the arc is, that is, the greater the curvature, so the curvature defined in this way is reasonable.

Generally, the bending degree of a curve is not equal everywhere, so it is meaningless to define the overall curvature, but it is meaningful to define the bending degree of a curve at a certain point. Obviously, it is natural and reasonable to define it as the curvature of the closed circle of the curve at this point.

So what is a closed circle? Let's first look at what the tangent is-the tangent is the limit chord. A chord is a line segment connecting two points on a curve. When two points are very close, it is the simplest approximation to replace the clamping curve with a chord. Let's improve this approximation: take three close points on the curve, make an arc (lower arc, of course) connecting these three points, and replace that curve with an arc, because the arc is a simple curve next to a straight line. When three points are infinitely close, the limit arc is obtained, and the limit circle drawn along this arc is the osculating circle.

The curvature of the curve defined in this way, expressed by differential formula, is of course da/ds, that is, the radian of bending per arc length.

Obviously, curvature is an intrinsic geometric quantity of the curve, that is, it has nothing to do with the selection of coordinate system, while da/dx is not. Obviously, it is a quantity relative to the coordinate system.