Mixing
How does twisting a patch of space changes its metric properties?
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So suppose I have a manifold $mathcalM$ that is embedded in the euclidian space $mathbbR^3$. $mathcalM$ is essentially a patch of the Euclidian space and it has well-defined boundaries. Like a cube of jelly floating in space.
Now, suppose I have a map $Phi:mathcalM rightarrow mathcalM$. $Phi$ is invertible and continuously differentiable, which means it is a diffeomorphism.
I am treating $Phi(mathcalM) = mathcalM'$ as the result from the application of $Phi$ collectively, at all points in $mathcalM$. I am thinking of this as if $mathcalM'$ is a "warped" version of $mathcalM$. Not sure if this is a valid way of considering the problem, but this way of framing it is necessary for the question.
I tried to numerically study the result of the application of various maps at my computer. The way I did this was by discretizing $mathcalM$ as a rectangular grid of finite size. I computed the connectivities of the points and created a mesh data structure that kept the information about the grid connectivities and vicinities of each point. Then, when I applied $Phi$, I had the information of what point used to be connected with what other point. This gave me a sense of how the distances had been distorted under $Phi$.
Fianlly, to really get a quantifiable sense of these distortions I wanted to compute the first and second fundamental forms at each point in $mathcalM'$ of my discretization, but in order to do that I needed a way to estimate the covariant basis vectors at each point in the distorted manifold. So I guess that my question is: Given this discretization scheme, how can I numerically estimate the covariant basis vectors at each point of my warped euclidian grid given the map $Phi$ that distorted it in the first place?
differential-geometry numerical-methods
asked Aug 3 at 23:17
urquiza
1338
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So suppose I have a manifold $mathcalM$ that is embedded in the euclidian space $mathbbR^3$. $mathcalM$ is essentially a patch of the Euclidian space and it has well-defined boundaries. Like a cube of jelly floating in space.
Now, suppose I have a map $Phi:mathcalM rightarrow mathcalM$. $Phi$ is invertible and continuously differentiable, which means it is a diffeomorphism.
I am treating $Phi(mathcalM) = mathcalM'$ as the result from the application of $Phi$ collectively, at all points in $mathcalM$. I am thinking of this as if $mathcalM'$ is a "warped" version of $mathcalM$. Not sure if this is a valid way of considering the problem, but this way of framing it is necessary for the question.
I tried to numerically study the result of the application of various maps at my computer. The way I did this was by discretizing $mathcalM$ as a rectangular grid of finite size. I computed the connectivities of the points and created a mesh data structure that kept the information about the grid connectivities and vicinities of each point. Then, when I applied $Phi$, I had the information of what point used to be connected with what other point. This gave me a sense of how the distances had been distorted under $Phi$.
Fianlly, to really get a quantifiable sense of these distortions I wanted to compute the first and second fundamental forms at each point in $mathcalM'$ of my discretization, but in order to do that I needed a way to estimate the covariant basis vectors at each point in the distorted manifold. So I guess that my question is: Given this discretization scheme, how can I numerically estimate the covariant basis vectors at each point of my warped euclidian grid given the map $Phi$ that distorted it in the first place?
differential-geometry numerical-methods
asked Aug 3 at 23:17
urquiza
1338
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
So suppose I have a manifold $mathcalM$ that is embedded in the euclidian space $mathbbR^3$. $mathcalM$ is essentially a patch of the Euclidian space and it has well-defined boundaries. Like a cube of jelly floating in space.
Now, suppose I have a map $Phi:mathcalM rightarrow mathcalM$. $Phi$ is invertible and continuously differentiable, which means it is a diffeomorphism.
I am treating $Phi(mathcalM) = mathcalM'$ as the result from the application of $Phi$ collectively, at all points in $mathcalM$. I am thinking of this as if $mathcalM'$ is a "warped" version of $mathcalM$. Not sure if this is a valid way of considering the problem, but this way of framing it is necessary for the question.
I tried to numerically study the result of the application of various maps at my computer. The way I did this was by discretizing $mathcalM$ as a rectangular grid of finite size. I computed the connectivities of the points and created a mesh data structure that kept the information about the grid connectivities and vicinities of each point. Then, when I applied $Phi$, I had the information of what point used to be connected with what other point. This gave me a sense of how the distances had been distorted under $Phi$.
Fianlly, to really get a quantifiable sense of these distortions I wanted to compute the first and second fundamental forms at each point in $mathcalM'$ of my discretization, but in order to do that I needed a way to estimate the covariant basis vectors at each point in the distorted manifold. So I guess that my question is: Given this discretization scheme, how can I numerically estimate the covariant basis vectors at each point of my warped euclidian grid given the map $Phi$ that distorted it in the first place?
differential-geometry numerical-methods
asked Aug 3 at 23:17
urquiza
1338
So suppose I have a manifold $mathcalM$ that is embedded in the euclidian space $mathbbR^3$. $mathcalM$ is essentially a patch of the Euclidian space and it has well-defined boundaries. Like a cube of jelly floating in space.
Now, suppose I have a map $Phi:mathcalM rightarrow mathcalM$. $Phi$ is invertible and continuously differentiable, which means it is a diffeomorphism.
I am treating $Phi(mathcalM) = mathcalM'$ as the result from the application of $Phi$ collectively, at all points in $mathcalM$. I am thinking of this as if $mathcalM'$ is a "warped" version of $mathcalM$. Not sure if this is a valid way of considering the problem, but this way of framing it is necessary for the question.
I tried to numerically study the result of the application of various maps at my computer. The way I did this was by discretizing $mathcalM$ as a rectangular grid of finite size. I computed the connectivities of the points and created a mesh data structure that kept the information about the grid connectivities and vicinities of each point. Then, when I applied $Phi$, I had the information of what point used to be connected with what other point. This gave me a sense of how the distances had been distorted under $Phi$.
Fianlly, to really get a quantifiable sense of these distortions I wanted to compute the first and second fundamental forms at each point in $mathcalM'$ of my discretization, but in order to do that I needed a way to estimate the covariant basis vectors at each point in the distorted manifold. So I guess that my question is: Given this discretization scheme, how can I numerically estimate the covariant basis vectors at each point of my warped euclidian grid given the map $Phi$ that distorted it in the first place?
differential-geometry numerical-methods
asked Aug 3 at 23:17
urquiza
1338
asked Aug 3 at 23:17
urquiza
1338
asked Aug 3 at 23:17
urquiza
1338
asked Aug 3 at 23:17
urquiza
1338
1338
add a comment |Â
add a comment |Â
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