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I'm asking this only because I could not find it anywhere on the web (tried Wikipedia and Google searches).

The only hint I found was in this image on Wikipedia, which seems to indicate that the radius of curvature is directed towards the centre of the osculating circle, which would mean the curvature vector itself is directed in the opposite direction.

But there's no clear definition anywhere. So: how is the direction of the curvature vector defined?

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    Naïvely, I would think the curvature vector should be the second derivative of the position vector with respect to arc length, which makes it point inwards.2012-10-10
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    I have the same question. Curvature is a scalar value that defines the rate at which a curve is changing direction with respect to arc length, but what is the vector that tells us the direction that this curve is changing with respect to arc length? My theory is that it is: [dx/ds,dy/ds,dz/ds]; however, I don't know how you'd compute a vector such as this. Tangent, normal, and binormal vectors do not point in the direction of curvature.2013-04-27
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    @Marcus -- the formula you gave will calculate the curve tangent. You need to differentiate again to get curvature.2013-04-28

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