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an upper-case , is the derivative of the unit tangent vector, , with respect to arc length, :

The curvature vector represents both the direction towards which the curve is turning as well as how sharply it turns.

The curvature vector has the following properties:
* The magnitude of the curvature vector is the curvature:

* The curvature vector is perpendicular to the unit tangent vector , or in terms of the dot product:
* The curvature vector is the second derivative of an arc-length parametrization :

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