Calculating torsion of an asymptotic curve with nonzero curvature
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I came across this problem when studying Gauss Map:
Show that $tau^2=-K$ on an asymptotic curve. Here $tau$ is torsion of the asymptotic curve $pmb r(u(s),v(s))$(with curvature $kappa$ nonzero) and $K$ is total curvature.
The only thing I can figure out is that since $dpmb{r}$ is an asymptotic direction then principal curvature $k_n=0$. Also I saw another exercise which says in such situation we have: $$tau=frac{1}{sqrt {EG-F^2}}
begin{vmatrix}(dot v)^2 & -dot udot v & (dot u)^2 \
E & F & G \
L & M & N \end{vmatrix}$$
This one I could figure out by calculation. But I can't see any connection between these two expressions.
I know this question has been asked here. But I haven't studied the theorem mentioned in that answer yet. Anyone could give another solution? Thanks in advance!
differential-geometry surfaces curves curvature
asked Nov 18 at 13:00
Greywhite
656
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up vote
0
down vote
favorite
I came across this problem when studying Gauss Map:
Show that $tau^2=-K$ on an asymptotic curve. Here $tau$ is torsion of the asymptotic curve $pmb r(u(s),v(s))$(with curvature $kappa$ nonzero) and $K$ is total curvature.
The only thing I can figure out is that since $dpmb{r}$ is an asymptotic direction then principal curvature $k_n=0$. Also I saw another exercise which says in such situation we have: $$tau=frac{1}{sqrt {EG-F^2}}
begin{vmatrix}(dot v)^2 & -dot udot v & (dot u)^2 \
E & F & G \
L & M & N \end{vmatrix}$$
This one I could figure out by calculation. But I can't see any connection between these two expressions.
I know this question has been asked here. But I haven't studied the theorem mentioned in that answer yet. Anyone could give another solution? Thanks in advance!
differential-geometry surfaces curves curvature
asked Nov 18 at 13:00
Greywhite
656
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I came across this problem when studying Gauss Map:
Show that $tau^2=-K$ on an asymptotic curve. Here $tau$ is torsion of the asymptotic curve $pmb r(u(s),v(s))$(with curvature $kappa$ nonzero) and $K$ is total curvature.
The only thing I can figure out is that since $dpmb{r}$ is an asymptotic direction then principal curvature $k_n=0$. Also I saw another exercise which says in such situation we have: $$tau=frac{1}{sqrt {EG-F^2}}
begin{vmatrix}(dot v)^2 & -dot udot v & (dot u)^2 \
E & F & G \
L & M & N \end{vmatrix}$$
This one I could figure out by calculation. But I can't see any connection between these two expressions.
I know this question has been asked here. But I haven't studied the theorem mentioned in that answer yet. Anyone could give another solution? Thanks in advance!
differential-geometry surfaces curves curvature
asked Nov 18 at 13:00
Greywhite
656
I came across this problem when studying Gauss Map:
Show that $tau^2=-K$ on an asymptotic curve. Here $tau$ is torsion of the asymptotic curve $pmb r(u(s),v(s))$(with curvature $kappa$ nonzero) and $K$ is total curvature.
The only thing I can figure out is that since $dpmb{r}$ is an asymptotic direction then principal curvature $k_n=0$. Also I saw another exercise which says in such situation we have: $$tau=frac{1}{sqrt {EG-F^2}}
begin{vmatrix}(dot v)^2 & -dot udot v & (dot u)^2 \
E & F & G \
L & M & N \end{vmatrix}$$
This one I could figure out by calculation. But I can't see any connection between these two expressions.
I know this question has been asked here. But I haven't studied the theorem mentioned in that answer yet. Anyone could give another solution? Thanks in advance!
differential-geometry surfaces curves curvature
differential-geometry surfaces curves curvature
asked Nov 18 at 13:00
Greywhite
656
asked Nov 18 at 13:00
Greywhite
656
asked Nov 18 at 13:00
Greywhite
656
asked Nov 18 at 13:00
Greywhite
656
asked Nov 18 at 13:00
Greywhite
656
656
add a comment |
add a comment |
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