Can I compute contour orientation without using polygon area sign?

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Most of the times, I determine contour orientation generating 2D points and computing the closed polygon area. Depending on the area value sign I can understand if the contour is oriented clockwise or not (see How to determine if a list of polygon points are in clockwise order?).

Would it be possible to do the same computations without generating 2D points? I mean, relying only on geometric curve properties?

We are interested in determining the orientation of contours like these ones without sampling them with 2D points.

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