# Geodesic curvature, normal curvature, and geodesic torsion

• Demon117
In summary: I agree with the sign issue but I think it does make a huge difference because N'\cdot n \ne N'\cdot N by definition of the Darboux frame, and by that same definition N'\cdot n=\tau_{g}. So II(T,N)\ne \tau_{g}.
Demon117
I am struggling to make sense out some things. Hopefully someone can help or at least offer some different point of view. Let's examine a differential curve parameterized by arc length that maps some interval into an oriented surface (lets call it N(s)). The surface has a unit normal field restricted to the curve $\alpha$. Also, let $n(s):=N(s)\wedge T(s)$ where $T(s)=\alpha '(s)$.

If we define the derivatives of T, N, and n as the following

$T'=-k_{g}n+k_{n}N$
$n'=k_{g}T+\tau_{g}N$
$N'=-k_{n}T-\tau_{g}n$

then we should have $N'\cdot T = -k_{n}$ and the second fundamental form is given by $II(T,T) = k_{n}$ while $N'\cdot n=-\tau_{g}$ so that the second fundamental form is given by $II(T,n)=\tau_{g}$.

This seems pretty clear to me, unless I have my definitions mixed up some how. Does this seem correct?

Demon117 said:
I am struggling to make sense out some things. Hopefully someone can help or at least offer some different point of view. Let's examine a differential curve parameterized by arc length that maps some interval into an oriented surface (lets call it N(s)). The surface has a unit normal field restricted to the curve $\alpha$. Also, let $n(s):=N(s)\wedge T(s)$ where $T(s)=\alpha '(s)$.

Hmm. I'm not sure what your ##N(s)## is supposed to be. Is ##N(s)## the normal of the surface at point ##\alpha(s)##?

If we define the derivatives of T, N, and n as the following

$T'=-k_{g}n+k_{n}N$
$n'=k_{g}T+\tau_{g}N$
$N'=-k_{n}T-\tau_{g}n$

I get some sign differences. I think you should see http://en.wikipedia.org/wiki/Darboux_frame for the correct definitions.

the second fundamental form is given by $II(T,T) = k_{n}$ while $N'\cdot n=-\tau_{g}$ so that the second fundamental form is given by $II(T,n)=\tau_{g}$.

Not sure where this comes from. Could you clarify?

micromass said:
Hmm. I'm not sure what your ##N(s)## is supposed to be. Is ##N(s)## the normal of the surface at point ##\alpha(s)##?

Yes that is correct. I made a mistake in my initial statement. N(s) is the unit normal field to the surface at $\alpha(s)$. Thanks for allowing me to clarify this. That means n(s) is the normal to the curve $\alpha$.

I get some sign differences. I think you should see http://en.wikipedia.org/wiki/Darboux_frame for the correct definitions.

I will look into this.

Not sure where this comes from. Could you clarify?

The second fundamental form defined by $II(v,v)=-\left\langle dN_{p}v,v\right\rangle$ for any v in the tangent plane at a point p of S.

Demon117 said:
The second fundamental form defined by $II(v,v)=-\left\langle dN_{p}v,v\right\rangle$ for any v in the tangent plane at a point p of S.

I agree with that then, up to signs. (Not that signs are all that important)

micromass said:
I agree with that then, up to signs. (Not that signs are all that important)

That is interesting, because I spoke with a professor recently about this and his claim was that the geodesic torsion was defined by $II(T,N)=\tau_{g}$. But by inspection this wouldn't make sense. I'm just confused a little I guess.

Demon117 said:
That is interesting, because I spoke with a professor recently about this and his claim was that the geodesic torsion was defined by $II(T,N)=\tau_{g}$. But by inspection this wouldn't make sense. I'm just confused a little I guess.

Signs are not so important anyway. If he defines his geodesic torsion like that, then I don't think it'll make a lot of difference. It'll yield the same theory up to sign.

micromass said:
Signs are not so important anyway. If he defines his geodesic torsion like that, then I don't think it'll make a lot of difference. It'll yield the same theory up to sign.

I agree with the sign issue but I think it does make a huge difference because $N'\cdot n \ne N'\cdot N$ by definition of the Darboux frame, and by that same definition $N'\cdot n=\tau_{g}$. So $II(T,N)\ne \tau_{g}$.

## 1. What is geodesic curvature?

Geodesic curvature is a measure of how much a curve deviates from a straight line on a curved surface. It is defined as the rate of change of the tangent vector along the curve with respect to the arc length parameter.

## 2. How is normal curvature calculated?

Normal curvature is calculated by finding the rate of change of the normal vector to the surface along a given curve. It is a measure of how much the surface curves in the direction perpendicular to the curve at a specific point.

## 3. What is the relationship between geodesic curvature and normal curvature?

Geodesic curvature and normal curvature are related through the Gauss equation, which states that the sum of the geodesic curvature and normal curvature at a point on a surface is equal to the total curvature of the surface at that point.

## 4. What is geodesic torsion?

Geodesic torsion is a measure of how much a curve twists as it moves along a surface. It is defined as the rate of change of the osculating plane along the curve with respect to the arc length parameter. In other words, it measures the deviation of the curve from being planar.

## 5. How are geodesic curvature, normal curvature, and geodesic torsion used in real-world applications?

Geodesic curvature, normal curvature, and geodesic torsion are important concepts in the study of differential geometry, and they have many applications in fields such as physics, engineering, and computer graphics. For example, they are used in the design of curved structures, the analysis of the behavior of particles moving on curved surfaces, and the creation of 3D models and animations.

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