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How does curvature describe the changing direction of a curve?

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differential geometry An annular strip (the region between two concentric circles) can be cut and bent into a helical strake that follows approximately the contour of a cylinder. Techniques of differential geometry are employed to find the dimensions of the annular strip that will best match the required curvature of the strake.

In differential geometry, curvature is defined as the rate at which the direction of a curve changes with respect to the distance traveled along that curve.1

Mathematically, this is expressed using the osculating circle—the circle that most closely approximates the curve at a specific point. If $r$ is the radius of this circle, the curvature is defined as the reciprocal, $\frac{1}{r}$ .12

This relationship describes the "changing direction" of a curve in the following ways:
* Rate of Change: As you move along a curve, the curvature measures the derivative of the tangent vector. A higher curvature indicates a sharper turn, meaning the direction is changing rapidly over a short distance.2
* Approximation: For a straight line, which does not change direction, the radius is considered infinite, resulting in a curvature of zero. Conversely, as a curve becomes tighter, the radius of the osculating circle decreases, causing the curvature value to increase.12

In essence, curvature quantifies how quickly a path deviates from a straight line at any given point.2