Normal Curvature Along a non arclength-parameterized curve
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I consider the surface $S=(x,y,z) in mathbbR^3: , z=x^2+y^2$. Clearly, it may be parameterizated by $phi(u,v)=(u,v,u^2+v^2)$. Now, I consider the curve $t to phi(t^2,t)$.
Which is the normal curvature along the previous curve? My problem is that the curve is not arclength-parametrized, so I cannot compute simply the second fundamental form in the point $phi(t^2,t)$. In this case, what can I do?
differential-geometry
asked Sep 8 at 9:32
TheWanderer
1,78811029
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up vote
1
down vote
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I consider the surface $S=(x,y,z) in mathbbR^3: , z=x^2+y^2$. Clearly, it may be parameterizated by $phi(u,v)=(u,v,u^2+v^2)$. Now, I consider the curve $t to phi(t^2,t)$.
Which is the normal curvature along the previous curve? My problem is that the curve is not arclength-parametrized, so I cannot compute simply the second fundamental form in the point $phi(t^2,t)$. In this case, what can I do?
differential-geometry
asked Sep 8 at 9:32
TheWanderer
1,78811029
Maybe you can use Meusnier Theorem since your curve and it's positive arc length parametrisation have same unit tangent vector field at same points? I might be wrong though.
– user3342072
Sep 8 at 11:52
Yes, I want to use Meusnier Theorem. Have I to compute the tangent vector $v$ to the curve, then to find its norm (in order to divide for it) and finally to compute the second fundamental form on $fracvVert v Vert$?
– TheWanderer
Sep 8 at 14:05
that's how I would have done it aswell, but Meusnier is valid only for all arc length parametrised, so there might be a problem. Atleast we can surely say that for every point of your curve (whenever it's regular) there exist an arclength parametrised curve with tangential $fracvVert vVert$ (as a normal section). And then use Meusnier. I think it's plausible.
– user3342072
Sep 8 at 14:16
Computations seems really cumbersome.
– TheWanderer
Sep 8 at 14:51
@user3342072 I think this is not the way to solve the exercise.
– TheWanderer
Sep 8 at 15:59
 |Â
show 1 more comment
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I consider the surface $S=(x,y,z) in mathbbR^3: , z=x^2+y^2$. Clearly, it may be parameterizated by $phi(u,v)=(u,v,u^2+v^2)$. Now, I consider the curve $t to phi(t^2,t)$.
Which is the normal curvature along the previous curve? My problem is that the curve is not arclength-parametrized, so I cannot compute simply the second fundamental form in the point $phi(t^2,t)$. In this case, what can I do?
differential-geometry
asked Sep 8 at 9:32
TheWanderer
1,78811029
I consider the surface $S=(x,y,z) in mathbbR^3: , z=x^2+y^2$. Clearly, it may be parameterizated by $phi(u,v)=(u,v,u^2+v^2)$. Now, I consider the curve $t to phi(t^2,t)$.
Which is the normal curvature along the previous curve? My problem is that the curve is not arclength-parametrized, so I cannot compute simply the second fundamental form in the point $phi(t^2,t)$. In this case, what can I do?
differential-geometry
differential-geometry
asked Sep 8 at 9:32
TheWanderer
1,78811029
asked Sep 8 at 9:32
TheWanderer
1,78811029
asked Sep 8 at 9:32
TheWanderer
1,78811029
asked Sep 8 at 9:32
TheWanderer
1,78811029
asked Sep 8 at 9:32
TheWanderer
1,78811029
1,78811029
Maybe you can use Meusnier Theorem since your curve and it's positive arc length parametrisation have same unit tangent vector field at same points? I might be wrong though.
– user3342072
Sep 8 at 11:52
Yes, I want to use Meusnier Theorem. Have I to compute the tangent vector $v$ to the curve, then to find its norm (in order to divide for it) and finally to compute the second fundamental form on $fracvVert v Vert$?
– TheWanderer
Sep 8 at 14:05
that's how I would have done it aswell, but Meusnier is valid only for all arc length parametrised, so there might be a problem. Atleast we can surely say that for every point of your curve (whenever it's regular) there exist an arclength parametrised curve with tangential $fracvVert vVert$ (as a normal section). And then use Meusnier. I think it's plausible.
– user3342072
Sep 8 at 14:16
Computations seems really cumbersome.
– TheWanderer
Sep 8 at 14:51
@user3342072 I think this is not the way to solve the exercise.
– TheWanderer
Sep 8 at 15:59
 |Â
show 1 more comment
Maybe you can use Meusnier Theorem since your curve and it's positive arc length parametrisation have same unit tangent vector field at same points? I might be wrong though.
– user3342072
Sep 8 at 11:52
Yes, I want to use Meusnier Theorem. Have I to compute the tangent vector $v$ to the curve, then to find its norm (in order to divide for it) and finally to compute the second fundamental form on $fracvVert v Vert$?
– TheWanderer
Sep 8 at 14:05
that's how I would have done it aswell, but Meusnier is valid only for all arc length parametrised, so there might be a problem. Atleast we can surely say that for every point of your curve (whenever it's regular) there exist an arclength parametrised curve with tangential $fracvVert vVert$ (as a normal section). And then use Meusnier. I think it's plausible.
– user3342072
Sep 8 at 14:16
Computations seems really cumbersome.
– TheWanderer
Sep 8 at 14:51
@user3342072 I think this is not the way to solve the exercise.
– TheWanderer
Sep 8 at 15:59
Maybe you can use Meusnier Theorem since your curve and it's positive arc length parametrisation have same unit tangent vector field at same points? I might be wrong though.
– user3342072
Sep 8 at 11:52
Maybe you can use Meusnier Theorem since your curve and it's positive arc length parametrisation have same unit tangent vector field at same points? I might be wrong though.
– user3342072
Sep 8 at 11:52
Yes, I want to use Meusnier Theorem. Have I to compute the tangent vector $v$ to the curve, then to find its norm (in order to divide for it) and finally to compute the second fundamental form on $fracvVert v Vert$?
– TheWanderer
Sep 8 at 14:05
Yes, I want to use Meusnier Theorem. Have I to compute the tangent vector $v$ to the curve, then to find its norm (in order to divide for it) and finally to compute the second fundamental form on $fracvVert v Vert$?
– TheWanderer
Sep 8 at 14:05
that's how I would have done it aswell, but Meusnier is valid only for all arc length parametrised, so there might be a problem. Atleast we can surely say that for every point of your curve (whenever it's regular) there exist an arclength parametrised curve with tangential $fracvVert vVert$ (as a normal section). And then use Meusnier. I think it's plausible.
– user3342072
Sep 8 at 14:16
that's how I would have done it aswell, but Meusnier is valid only for all arc length parametrised, so there might be a problem. Atleast we can surely say that for every point of your curve (whenever it's regular) there exist an arclength parametrised curve with tangential $fracvVert vVert$ (as a normal section). And then use Meusnier. I think it's plausible.
– user3342072
Sep 8 at 14:16
Computations seems really cumbersome.
– TheWanderer
Sep 8 at 14:51
Computations seems really cumbersome.
– TheWanderer
Sep 8 at 14:51
@user3342072 I think this is not the way to solve the exercise.
– TheWanderer
Sep 8 at 15:59
@user3342072 I think this is not the way to solve the exercise.
– TheWanderer
Sep 8 at 15:59
 |Â
show 1 more comment
1 Answer
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You can still compute curvature of a non-arclength-parametrized curve; there are various ways to do it. My favorite is just to adjust the usual calculations with the chain rule. See, for example, p. 13 of my differential geometry text. So compute $kappavec N$ at the point and then dot with the unit surface normal. (This is the Meusnier's formula application comments referred to.)
Alternatively, yes, evaluate the second fundamental form of the surface at $phi(t^2,t)$ on the unit tangent vector of the curve. At $t=0$ this will be easy enough; for other values of $t$ it will be yucky algebra either way, I guess.
I can answer further questions if you ask.
answered Sep 8 at 23:35
Ted Shifrin
60.9k44388
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
You can still compute curvature of a non-arclength-parametrized curve; there are various ways to do it. My favorite is just to adjust the usual calculations with the chain rule. See, for example, p. 13 of my differential geometry text. So compute $kappavec N$ at the point and then dot with the unit surface normal. (This is the Meusnier's formula application comments referred to.)
Alternatively, yes, evaluate the second fundamental form of the surface at $phi(t^2,t)$ on the unit tangent vector of the curve. At $t=0$ this will be easy enough; for other values of $t$ it will be yucky algebra either way, I guess.
I can answer further questions if you ask.
answered Sep 8 at 23:35
Ted Shifrin
60.9k44388
add a comment |Â
up vote
2
down vote
accepted
You can still compute curvature of a non-arclength-parametrized curve; there are various ways to do it. My favorite is just to adjust the usual calculations with the chain rule. See, for example, p. 13 of my differential geometry text. So compute $kappavec N$ at the point and then dot with the unit surface normal. (This is the Meusnier's formula application comments referred to.)
Alternatively, yes, evaluate the second fundamental form of the surface at $phi(t^2,t)$ on the unit tangent vector of the curve. At $t=0$ this will be easy enough; for other values of $t$ it will be yucky algebra either way, I guess.
I can answer further questions if you ask.
answered Sep 8 at 23:35
Ted Shifrin
60.9k44388
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
You can still compute curvature of a non-arclength-parametrized curve; there are various ways to do it. My favorite is just to adjust the usual calculations with the chain rule. See, for example, p. 13 of my differential geometry text. So compute $kappavec N$ at the point and then dot with the unit surface normal. (This is the Meusnier's formula application comments referred to.)
Alternatively, yes, evaluate the second fundamental form of the surface at $phi(t^2,t)$ on the unit tangent vector of the curve. At $t=0$ this will be easy enough; for other values of $t$ it will be yucky algebra either way, I guess.
I can answer further questions if you ask.
answered Sep 8 at 23:35
Ted Shifrin
60.9k44388
You can still compute curvature of a non-arclength-parametrized curve; there are various ways to do it. My favorite is just to adjust the usual calculations with the chain rule. See, for example, p. 13 of my differential geometry text. So compute $kappavec N$ at the point and then dot with the unit surface normal. (This is the Meusnier's formula application comments referred to.)
Alternatively, yes, evaluate the second fundamental form of the surface at $phi(t^2,t)$ on the unit tangent vector of the curve. At $t=0$ this will be easy enough; for other values of $t$ it will be yucky algebra either way, I guess.
I can answer further questions if you ask.
answered Sep 8 at 23:35
Ted Shifrin
60.9k44388
answered Sep 8 at 23:35
Ted Shifrin
60.9k44388
answered Sep 8 at 23:35
Ted Shifrin
60.9k44388
answered Sep 8 at 23:35
Ted Shifrin
60.9k44388
60.9k44388
add a comment |Â
add a comment |Â
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Maybe you can use Meusnier Theorem since your curve and it's positive arc length parametrisation have same unit tangent vector field at same points? I might be wrong though.
– user3342072
Sep 8 at 11:52
Yes, I want to use Meusnier Theorem. Have I to compute the tangent vector $v$ to the curve, then to find its norm (in order to divide for it) and finally to compute the second fundamental form on $fracvVert v Vert$?
– TheWanderer
Sep 8 at 14:05
that's how I would have done it aswell, but Meusnier is valid only for all arc length parametrised, so there might be a problem. Atleast we can surely say that for every point of your curve (whenever it's regular) there exist an arclength parametrised curve with tangential $fracvVert vVert$ (as a normal section). And then use Meusnier. I think it's plausible.
– user3342072
Sep 8 at 14:16
Computations seems really cumbersome.
– TheWanderer
Sep 8 at 14:51
@user3342072 I think this is not the way to solve the exercise.
– TheWanderer
Sep 8 at 15:59