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.

 

[blue_raver22]

Yes, it is possible to define an angle that represents torsion. This angle is known as the second tangential angle or the torsion angle. It is defined as the angle between the tangent vector and the binormal vector in a Frenet frame at a given point on a curve. This angle is important in understanding the three-dimensional curvature of a curve, as it measures the rate of change of the direction of the tangent vector with respect to arc length.

The torsion angle is derived by taking the derivative of the tangent vector with respect to arc length, and then taking the dot product of this derivative with the binormal vector. This dot product can also be represented as the cross product of the tangent and binormal vectors, which is why the torsion angle is also sometimes referred to as the "twist" of a curve.

Torsion plays a crucial role in many areas of mathematics and physics, including differential geometry, mechanics, and fluid dynamics. It is particularly important in the study of curves and surfaces in three-dimensional space.

In conclusion, the second tangential angle, or torsion angle, is a well-defined concept that measures the rate of change of the tangent vector with respect to arc length. It provides valuable insight into the three-dimensional curvature of a curve and is essential in many fields of science and mathematics.