2nd tangential angle and torsion

In summary, the conversation discusses the concept of a second tangential angle and its relationship to torsion in curves. It is mentioned that the second tangential angle is only defined for plane curves and that for space curves, the tangents are not in the same plane. The possibility of defining the second tangential angle as the angle between osculating planes is also discussed, but it is suggested that it is easier to study the angle between the binormals instead.
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  • #2
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.
 
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  • #3
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?

asd.jpg
 
  • #4
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.
 
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  • #5


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.
 

Related to 2nd tangential angle and torsion

What is a 2nd tangential angle?

A 2nd tangential angle is an angle that is formed between two tangents of a curve at a specific point. It is also known as the "2nd derivative of the curve at that point."

What is torsion in relation to 2nd tangential angle?

Torsion is a measure of how much a curve in three-dimensional space is twisting at a specific point. It is related to the 2nd tangential angle because the 2nd derivative of a curve can be used to calculate the torsion at a particular point.

How is the 2nd tangential angle calculated?

The 2nd tangential angle can be calculated by taking the second derivative of the curve at a specific point. This involves taking the derivative of the slope of the curve at that point, or in other words, the rate of change of the first derivative of the curve.

What is the significance of the 2nd tangential angle in mathematics?

The 2nd tangential angle is an important concept in differential geometry and calculus. It is used to understand the curvature and torsion of curves in three-dimensional space, and has applications in fields such as physics, engineering, and computer graphics.

Can the 2nd tangential angle and torsion be negative?

Yes, the 2nd tangential angle and torsion can both be negative. Negative values indicate that the curve is twisting in the opposite direction (clockwise) compared to positive values (counterclockwise).

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