2nd tangential angle and torsion

[Jhenrique]

"2nd tangential angle" and torsion

When I derive dθ/ds I get the curvature k of a curve. But exist too the torsion τ of a curve and I think that exist some angle that when derivate wrt arc length s results in the torsion τ. So, is possible to define such angle?

http://mathworld.wolfram.com/TangentialAngle.html
http://mathworld.wolfram.com/FrenetFormulas.html

 

[Abel Cavaşi]

Tangential angle is defined only for plane curves. A curve that have torsion is not a plane curve. So, for a space curve the tangents are not in the same plane. Optionally, you can define "second tangential angle" just the angle between osculating planes.

 

[Jhenrique]

But if I to project the tangent vector in the xy plane thus we can see the tangential angle (in blue) and the red angle would be the tangle that when derivate wrt to arc length s results the torsion. This scheme is valid?

 

[Abel Cavaşi]

I'm afraid that you have entered into too much complication. It seems to me that everything is easier. Whereas we are interested in the angle between the osculating planes and knowing that binormal vector is perpendicular to osculating plane, it is sufficient to study only the angle between the binormals. Thus, the sought angle is the angle between the binormals. So, I think that the drawing is not correct in this way.