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Does “isoparametric†mean what it sounds like?
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Please forgive my inexact language.
Given an infinitely dense family of smooth curves on $mathbbR^2$, all of which are parameterized by the same variable such that points with the same parameter value on "adjacent" curves are "contiguous". I want to call the entire set of points determined by a given parameter value an isoparametric curve.
For example, imagine some well behaved 2-dimensional fluid flow. At time $t=0$ a straight line segment of dye marker is emitted roughly perpendicular to the fluid flow, thereby designating those fluid points as having the same time parameter value. The time evolution of that line segment would constitute an isoparametric curve.
Is this a proper use of the term isoparametric?
riemannian-geometry vector-analysis smooth-manifolds
asked Jul 29 at 19:21
Steven Hatton
631314
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up vote
0
down vote
favorite
Please forgive my inexact language.
Given an infinitely dense family of smooth curves on $mathbbR^2$, all of which are parameterized by the same variable such that points with the same parameter value on "adjacent" curves are "contiguous". I want to call the entire set of points determined by a given parameter value an isoparametric curve.
For example, imagine some well behaved 2-dimensional fluid flow. At time $t=0$ a straight line segment of dye marker is emitted roughly perpendicular to the fluid flow, thereby designating those fluid points as having the same time parameter value. The time evolution of that line segment would constitute an isoparametric curve.
Is this a proper use of the term isoparametric?
riemannian-geometry vector-analysis smooth-manifolds
asked Jul 29 at 19:21
Steven Hatton
631314
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Please forgive my inexact language.
Given an infinitely dense family of smooth curves on $mathbbR^2$, all of which are parameterized by the same variable such that points with the same parameter value on "adjacent" curves are "contiguous". I want to call the entire set of points determined by a given parameter value an isoparametric curve.
For example, imagine some well behaved 2-dimensional fluid flow. At time $t=0$ a straight line segment of dye marker is emitted roughly perpendicular to the fluid flow, thereby designating those fluid points as having the same time parameter value. The time evolution of that line segment would constitute an isoparametric curve.
Is this a proper use of the term isoparametric?
riemannian-geometry vector-analysis smooth-manifolds
asked Jul 29 at 19:21
Steven Hatton
631314
Please forgive my inexact language.
Given an infinitely dense family of smooth curves on $mathbbR^2$, all of which are parameterized by the same variable such that points with the same parameter value on "adjacent" curves are "contiguous". I want to call the entire set of points determined by a given parameter value an isoparametric curve.
For example, imagine some well behaved 2-dimensional fluid flow. At time $t=0$ a straight line segment of dye marker is emitted roughly perpendicular to the fluid flow, thereby designating those fluid points as having the same time parameter value. The time evolution of that line segment would constitute an isoparametric curve.
Is this a proper use of the term isoparametric?
riemannian-geometry vector-analysis smooth-manifolds
asked Jul 29 at 19:21
Steven Hatton
631314
asked Jul 29 at 19:21
Steven Hatton
631314
asked Jul 29 at 19:21
Steven Hatton
631314
asked Jul 29 at 19:21
Steven Hatton
631314
631314
add a comment |Â
add a comment |Â
2 Answers
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Yes, that usage is reasonable: lines of constant parameter value on tensor-product subdivision surfaces are commonly called "isoparametric lines," for instance.
Note that there is at least a potential for confusion with the similar-sounding "isoperimetric," so I would avoid "isoparametric" if you are also solving variational problems using your family of curves.
answered Jul 29 at 19:29
user7530
33.3k558109
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Beware, the word isoparametric is used in differential geometry in a different context (though I think it's not very extended).
A smooth function $f:mathbbR^ntomathbbR$ (usually required to be non-constant in every open subset of $mathbbR^n$) is said to be isoparametric if
$Delta f=acirc f$ and $|nabla f|^2=bcirc f$ for $ain C^0(mathbbR), bin C^2(mathbbR)$.
An isoparametric manifold is a regular level set of an isoparametric function.
The name is due to historical reasons. Here's a reference to this theory.
answered Jul 29 at 19:37
Javi
2,1481725
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Yes, that usage is reasonable: lines of constant parameter value on tensor-product subdivision surfaces are commonly called "isoparametric lines," for instance.
Note that there is at least a potential for confusion with the similar-sounding "isoperimetric," so I would avoid "isoparametric" if you are also solving variational problems using your family of curves.
answered Jul 29 at 19:29
user7530
33.3k558109
add a comment |Â
up vote
1
down vote
accepted
Yes, that usage is reasonable: lines of constant parameter value on tensor-product subdivision surfaces are commonly called "isoparametric lines," for instance.
Note that there is at least a potential for confusion with the similar-sounding "isoperimetric," so I would avoid "isoparametric" if you are also solving variational problems using your family of curves.
answered Jul 29 at 19:29
user7530
33.3k558109
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Yes, that usage is reasonable: lines of constant parameter value on tensor-product subdivision surfaces are commonly called "isoparametric lines," for instance.
Note that there is at least a potential for confusion with the similar-sounding "isoperimetric," so I would avoid "isoparametric" if you are also solving variational problems using your family of curves.
answered Jul 29 at 19:29
user7530
33.3k558109
Yes, that usage is reasonable: lines of constant parameter value on tensor-product subdivision surfaces are commonly called "isoparametric lines," for instance.
Note that there is at least a potential for confusion with the similar-sounding "isoperimetric," so I would avoid "isoparametric" if you are also solving variational problems using your family of curves.
answered Jul 29 at 19:29
user7530
33.3k558109
answered Jul 29 at 19:29
user7530
33.3k558109
answered Jul 29 at 19:29
user7530
33.3k558109
answered Jul 29 at 19:29
user7530
33.3k558109
33.3k558109
add a comment |Â
add a comment |Â
up vote
1
down vote
Beware, the word isoparametric is used in differential geometry in a different context (though I think it's not very extended).
A smooth function $f:mathbbR^ntomathbbR$ (usually required to be non-constant in every open subset of $mathbbR^n$) is said to be isoparametric if
$Delta f=acirc f$ and $|nabla f|^2=bcirc f$ for $ain C^0(mathbbR), bin C^2(mathbbR)$.
An isoparametric manifold is a regular level set of an isoparametric function.
The name is due to historical reasons. Here's a reference to this theory.
answered Jul 29 at 19:37
Javi
2,1481725
add a comment |Â
up vote
1
down vote
Beware, the word isoparametric is used in differential geometry in a different context (though I think it's not very extended).
A smooth function $f:mathbbR^ntomathbbR$ (usually required to be non-constant in every open subset of $mathbbR^n$) is said to be isoparametric if
$Delta f=acirc f$ and $|nabla f|^2=bcirc f$ for $ain C^0(mathbbR), bin C^2(mathbbR)$.
An isoparametric manifold is a regular level set of an isoparametric function.
The name is due to historical reasons. Here's a reference to this theory.
answered Jul 29 at 19:37
Javi
2,1481725
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Beware, the word isoparametric is used in differential geometry in a different context (though I think it's not very extended).
A smooth function $f:mathbbR^ntomathbbR$ (usually required to be non-constant in every open subset of $mathbbR^n$) is said to be isoparametric if
$Delta f=acirc f$ and $|nabla f|^2=bcirc f$ for $ain C^0(mathbbR), bin C^2(mathbbR)$.
An isoparametric manifold is a regular level set of an isoparametric function.
The name is due to historical reasons. Here's a reference to this theory.
answered Jul 29 at 19:37
Javi
2,1481725
Beware, the word isoparametric is used in differential geometry in a different context (though I think it's not very extended).
A smooth function $f:mathbbR^ntomathbbR$ (usually required to be non-constant in every open subset of $mathbbR^n$) is said to be isoparametric if
$Delta f=acirc f$ and $|nabla f|^2=bcirc f$ for $ain C^0(mathbbR), bin C^2(mathbbR)$.
An isoparametric manifold is a regular level set of an isoparametric function.
The name is due to historical reasons. Here's a reference to this theory.
answered Jul 29 at 19:37
Javi
2,1481725
answered Jul 29 at 19:37
Javi
2,1481725
answered Jul 29 at 19:37
Javi
2,1481725
answered Jul 29 at 19:37
Javi
2,1481725
2,1481725
add a comment |Â
add a comment |Â
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