Dividing a closed connected region into multiple subregions
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Suppose I have a closed connected bounded region $A$ in $R^2$ (I am not that familiar with definitions in topology, so I just use the word "region" to mean the obvious one). It's probably called a closed connected bounded set in $R^2$, is it not?
If I want to partition $A$ into $n+1$ subregions with equal area using $n$ vertical lines, where $n$ is even, how do I do that?
I mean, I am sure it can be done, but I'd like to find these exact lines. My first guess is I have to find the parametrization $(x(t),y(t))$ of the boundary, but I don't know where to go next, but my next guess is to use the concept of line integral.
I assume even $n$ since I guess it is a bit more complicated than the odd ones. Just maybe..
*feel free to change the tags
multivariable-calculus analytic-geometry
asked Jan 2 at 14:31
bmsbms
335
$endgroup$
add a comment |
$begingroup$
Suppose I have a closed connected bounded region $A$ in $R^2$ (I am not that familiar with definitions in topology, so I just use the word "region" to mean the obvious one). It's probably called a closed connected bounded set in $R^2$, is it not?
If I want to partition $A$ into $n+1$ subregions with equal area using $n$ vertical lines, where $n$ is even, how do I do that?
I mean, I am sure it can be done, but I'd like to find these exact lines. My first guess is I have to find the parametrization $(x(t),y(t))$ of the boundary, but I don't know where to go next, but my next guess is to use the concept of line integral.
I assume even $n$ since I guess it is a bit more complicated than the odd ones. Just maybe..
*feel free to change the tags
multivariable-calculus analytic-geometry
asked Jan 2 at 14:31
bmsbms
335
$endgroup$
$begingroup$
You should add some more details: do you know an equation for the boundary in the form $y=f(x)$, with $f$ regular enough? is the region convex? do you need an exact result, or just a numeric approximation? A realistic example would also be of help.
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– Aretino
Jan 2 at 16:04
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@Aretino I did not know anything about convex until you mentioned it and went on to read what it is. The region need not be convex: two arbitrary points in it are connected by a curve (not necessarily straight line) in it. If the region is closed (like a disk for example), I thought the general equation needs parametrization, not just y = f(x)? I think we can assume that the parametrization is known.
$endgroup$
– bms
Jan 3 at 0:09
add a comment |
$begingroup$
Suppose I have a closed connected bounded region $A$ in $R^2$ (I am not that familiar with definitions in topology, so I just use the word "region" to mean the obvious one). It's probably called a closed connected bounded set in $R^2$, is it not?
If I want to partition $A$ into $n+1$ subregions with equal area using $n$ vertical lines, where $n$ is even, how do I do that?
I mean, I am sure it can be done, but I'd like to find these exact lines. My first guess is I have to find the parametrization $(x(t),y(t))$ of the boundary, but I don't know where to go next, but my next guess is to use the concept of line integral.
I assume even $n$ since I guess it is a bit more complicated than the odd ones. Just maybe..
*feel free to change the tags
multivariable-calculus analytic-geometry
asked Jan 2 at 14:31
bmsbms
335
$endgroup$
Suppose I have a closed connected bounded region $A$ in $R^2$ (I am not that familiar with definitions in topology, so I just use the word "region" to mean the obvious one). It's probably called a closed connected bounded set in $R^2$, is it not?
If I want to partition $A$ into $n+1$ subregions with equal area using $n$ vertical lines, where $n$ is even, how do I do that?
I mean, I am sure it can be done, but I'd like to find these exact lines. My first guess is I have to find the parametrization $(x(t),y(t))$ of the boundary, but I don't know where to go next, but my next guess is to use the concept of line integral.
I assume even $n$ since I guess it is a bit more complicated than the odd ones. Just maybe..
*feel free to change the tags
multivariable-calculus analytic-geometry
multivariable-calculus analytic-geometry
asked Jan 2 at 14:31
bmsbms
335
asked Jan 2 at 14:31
bmsbms
335
asked Jan 2 at 14:31
bmsbms
335
asked Jan 2 at 14:31
bmsbms
335
asked Jan 2 at 14:31
bmsbms
335
335
$begingroup$
You should add some more details: do you know an equation for the boundary in the form $y=f(x)$, with $f$ regular enough? is the region convex? do you need an exact result, or just a numeric approximation? A realistic example would also be of help.
$endgroup$
– Aretino
Jan 2 at 16:04
$begingroup$
@Aretino I did not know anything about convex until you mentioned it and went on to read what it is. The region need not be convex: two arbitrary points in it are connected by a curve (not necessarily straight line) in it. If the region is closed (like a disk for example), I thought the general equation needs parametrization, not just y = f(x)? I think we can assume that the parametrization is known.
$endgroup$
– bms
Jan 3 at 0:09
add a comment |
$begingroup$
You should add some more details: do you know an equation for the boundary in the form $y=f(x)$, with $f$ regular enough? is the region convex? do you need an exact result, or just a numeric approximation? A realistic example would also be of help.
$endgroup$
– Aretino
Jan 2 at 16:04
$begingroup$
@Aretino I did not know anything about convex until you mentioned it and went on to read what it is. The region need not be convex: two arbitrary points in it are connected by a curve (not necessarily straight line) in it. If the region is closed (like a disk for example), I thought the general equation needs parametrization, not just y = f(x)? I think we can assume that the parametrization is known.
$endgroup$
– bms
Jan 3 at 0:09
$begingroup$
You should add some more details: do you know an equation for the boundary in the form $y=f(x)$, with $f$ regular enough? is the region convex? do you need an exact result, or just a numeric approximation? A realistic example would also be of help.
$endgroup$
– Aretino
Jan 2 at 16:04
$begingroup$
You should add some more details: do you know an equation for the boundary in the form $y=f(x)$, with $f$ regular enough? is the region convex? do you need an exact result, or just a numeric approximation? A realistic example would also be of help.
$endgroup$
– Aretino
Jan 2 at 16:04
$begingroup$
@Aretino I did not know anything about convex until you mentioned it and went on to read what it is. The region need not be convex: two arbitrary points in it are connected by a curve (not necessarily straight line) in it. If the region is closed (like a disk for example), I thought the general equation needs parametrization, not just y = f(x)? I think we can assume that the parametrization is known.
$endgroup$
– bms
Jan 3 at 0:09
$begingroup$
@Aretino I did not know anything about convex until you mentioned it and went on to read what it is. The region need not be convex: two arbitrary points in it are connected by a curve (not necessarily straight line) in it. If the region is closed (like a disk for example), I thought the general equation needs parametrization, not just y = f(x)? I think we can assume that the parametrization is known.
$endgroup$
– bms
Jan 3 at 0:09
add a comment |
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$begingroup$
You should add some more details: do you know an equation for the boundary in the form $y=f(x)$, with $f$ regular enough? is the region convex? do you need an exact result, or just a numeric approximation? A realistic example would also be of help.
$endgroup$
– Aretino
Jan 2 at 16:04
$begingroup$
@Aretino I did not know anything about convex until you mentioned it and went on to read what it is. The region need not be convex: two arbitrary points in it are connected by a curve (not necessarily straight line) in it. If the region is closed (like a disk for example), I thought the general equation needs parametrization, not just y = f(x)? I think we can assume that the parametrization is known.
$endgroup$
– bms
Jan 3 at 0:09