Are planes one-sided or two-sided?
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I'm looking for clarity here - because, although it seems that planes are treated normally as one-sided, as I understand it, when I read of 'projection onto a plane', it seems that the projection is going to be on a single side of the plane. But, for instance, if we were to use both sides of the plane, then a spherical projection onto a plane could use 'the other side' for the upper hemisphere, but it doesn't seem to be done that way?
A simple, but well-illustrated and intuitive answer would be welcome.
geometry projection
asked 22 hours ago
Konchog
1112
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I'm looking for clarity here - because, although it seems that planes are treated normally as one-sided, as I understand it, when I read of 'projection onto a plane', it seems that the projection is going to be on a single side of the plane. But, for instance, if we were to use both sides of the plane, then a spherical projection onto a plane could use 'the other side' for the upper hemisphere, but it doesn't seem to be done that way?
A simple, but well-illustrated and intuitive answer would be welcome.
geometry projection
asked 22 hours ago
Konchog
1112
there is no sides of planes in the standard geometry. A projection into a plane is part of the plane, so it have "two sides" or "one side", whatever you want to consider
– Masacroso
22 hours ago
A plane has no sides, conventionally. The space around the plane, however, does. And there are applications where surfaces are considered to have sides. See, for instance, this video about turning spheres inside-out. Here the sphere surface does have two sides (a yellow and a purple), and there is nothing stopping you from making a plane from the same "material".
– Arthur
22 hours ago
Thanks - I read: "A plane is a flat, two-dimensional surface." As a 'muggle', I find it difficult to comprehend the notion of a surface that has no sides. It's why I asked for some sort of illustration rather than merely a statement of fact.
– Konchog
22 hours ago
@Konchog I think you would need to define/describe what you mean by 'two-sided'. 'Visually' it makes sense if I think of a sheet of paper for example. However even thinking of the projection you describe, it does not 'need' two sides: A vector/point in the plane is just a vector/point in the plane.
– AnyAD
21 hours ago
A projection of some $Ssubset Bbb R^3$ to a plane $Psubset Bbb R^3$ maps each point of $S$ to a point of $P.$ And a point in $P$ does not have a "side". It does not belong to the "top" or "bottom" of $P.$ In everyday physical reality we do not encounter infinitely thin objects with zero volume.
– DanielWainfleet
20 hours ago
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up vote
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down vote
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I'm looking for clarity here - because, although it seems that planes are treated normally as one-sided, as I understand it, when I read of 'projection onto a plane', it seems that the projection is going to be on a single side of the plane. But, for instance, if we were to use both sides of the plane, then a spherical projection onto a plane could use 'the other side' for the upper hemisphere, but it doesn't seem to be done that way?
A simple, but well-illustrated and intuitive answer would be welcome.
geometry projection
asked 22 hours ago
Konchog
1112
I'm looking for clarity here - because, although it seems that planes are treated normally as one-sided, as I understand it, when I read of 'projection onto a plane', it seems that the projection is going to be on a single side of the plane. But, for instance, if we were to use both sides of the plane, then a spherical projection onto a plane could use 'the other side' for the upper hemisphere, but it doesn't seem to be done that way?
A simple, but well-illustrated and intuitive answer would be welcome.
geometry projection
geometry projection
asked 22 hours ago
Konchog
1112
asked 22 hours ago
Konchog
1112
asked 22 hours ago
Konchog
1112
asked 22 hours ago
Konchog
1112
asked 22 hours ago
Konchog
1112
1112
there is no sides of planes in the standard geometry. A projection into a plane is part of the plane, so it have "two sides" or "one side", whatever you want to consider
– Masacroso
22 hours ago
A plane has no sides, conventionally. The space around the plane, however, does. And there are applications where surfaces are considered to have sides. See, for instance, this video about turning spheres inside-out. Here the sphere surface does have two sides (a yellow and a purple), and there is nothing stopping you from making a plane from the same "material".
– Arthur
22 hours ago
Thanks - I read: "A plane is a flat, two-dimensional surface." As a 'muggle', I find it difficult to comprehend the notion of a surface that has no sides. It's why I asked for some sort of illustration rather than merely a statement of fact.
– Konchog
22 hours ago
@Konchog I think you would need to define/describe what you mean by 'two-sided'. 'Visually' it makes sense if I think of a sheet of paper for example. However even thinking of the projection you describe, it does not 'need' two sides: A vector/point in the plane is just a vector/point in the plane.
– AnyAD
21 hours ago
A projection of some $Ssubset Bbb R^3$ to a plane $Psubset Bbb R^3$ maps each point of $S$ to a point of $P.$ And a point in $P$ does not have a "side". It does not belong to the "top" or "bottom" of $P.$ In everyday physical reality we do not encounter infinitely thin objects with zero volume.
– DanielWainfleet
20 hours ago
add a comment |
there is no sides of planes in the standard geometry. A projection into a plane is part of the plane, so it have "two sides" or "one side", whatever you want to consider
– Masacroso
22 hours ago
A plane has no sides, conventionally. The space around the plane, however, does. And there are applications where surfaces are considered to have sides. See, for instance, this video about turning spheres inside-out. Here the sphere surface does have two sides (a yellow and a purple), and there is nothing stopping you from making a plane from the same "material".
– Arthur
22 hours ago
Thanks - I read: "A plane is a flat, two-dimensional surface." As a 'muggle', I find it difficult to comprehend the notion of a surface that has no sides. It's why I asked for some sort of illustration rather than merely a statement of fact.
– Konchog
22 hours ago
@Konchog I think you would need to define/describe what you mean by 'two-sided'. 'Visually' it makes sense if I think of a sheet of paper for example. However even thinking of the projection you describe, it does not 'need' two sides: A vector/point in the plane is just a vector/point in the plane.
– AnyAD
21 hours ago
A projection of some $Ssubset Bbb R^3$ to a plane $Psubset Bbb R^3$ maps each point of $S$ to a point of $P.$ And a point in $P$ does not have a "side". It does not belong to the "top" or "bottom" of $P.$ In everyday physical reality we do not encounter infinitely thin objects with zero volume.
– DanielWainfleet
20 hours ago
there is no sides of planes in the standard geometry. A projection into a plane is part of the plane, so it have "two sides" or "one side", whatever you want to consider
– Masacroso
22 hours ago
there is no sides of planes in the standard geometry. A projection into a plane is part of the plane, so it have "two sides" or "one side", whatever you want to consider
– Masacroso
22 hours ago
A plane has no sides, conventionally. The space around the plane, however, does. And there are applications where surfaces are considered to have sides. See, for instance, this video about turning spheres inside-out. Here the sphere surface does have two sides (a yellow and a purple), and there is nothing stopping you from making a plane from the same "material".
– Arthur
22 hours ago
A plane has no sides, conventionally. The space around the plane, however, does. And there are applications where surfaces are considered to have sides. See, for instance, this video about turning spheres inside-out. Here the sphere surface does have two sides (a yellow and a purple), and there is nothing stopping you from making a plane from the same "material".
– Arthur
22 hours ago
Thanks - I read: "A plane is a flat, two-dimensional surface." As a 'muggle', I find it difficult to comprehend the notion of a surface that has no sides. It's why I asked for some sort of illustration rather than merely a statement of fact.
– Konchog
22 hours ago
Thanks - I read: "A plane is a flat, two-dimensional surface." As a 'muggle', I find it difficult to comprehend the notion of a surface that has no sides. It's why I asked for some sort of illustration rather than merely a statement of fact.
– Konchog
22 hours ago
@Konchog I think you would need to define/describe what you mean by 'two-sided'. 'Visually' it makes sense if I think of a sheet of paper for example. However even thinking of the projection you describe, it does not 'need' two sides: A vector/point in the plane is just a vector/point in the plane.
– AnyAD
21 hours ago
@Konchog I think you would need to define/describe what you mean by 'two-sided'. 'Visually' it makes sense if I think of a sheet of paper for example. However even thinking of the projection you describe, it does not 'need' two sides: A vector/point in the plane is just a vector/point in the plane.
– AnyAD
21 hours ago
A projection of some $Ssubset Bbb R^3$ to a plane $Psubset Bbb R^3$ maps each point of $S$ to a point of $P.$ And a point in $P$ does not have a "side". It does not belong to the "top" or "bottom" of $P.$ In everyday physical reality we do not encounter infinitely thin objects with zero volume.
– DanielWainfleet
20 hours ago
A projection of some $Ssubset Bbb R^3$ to a plane $Psubset Bbb R^3$ maps each point of $S$ to a point of $P.$ And a point in $P$ does not have a "side". It does not belong to the "top" or "bottom" of $P.$ In everyday physical reality we do not encounter infinitely thin objects with zero volume.
– DanielWainfleet
20 hours ago
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It depends on what you consider two-sided. Let's think of your simple example of the surface of the unit sphere in $mathbb R^3$. We fix the norh pole $x=(0,0,1)$.
If you want to move from $x$ out of the plane, you can move either inside or outside the sphere. This is due to the fact, that there exist two possible normal vectors. The distinction of these normal vectors is important for example in Gauss' Thorem.
If you think of sides similar to sides of a cube i.e. surfaces that enclose your set, or boundaries, then a plane consists only of one boundary. If you consider again the north pose $x$ it does not lie on the inside or outside of the surface of the sphere, and the same holds for every point on any plane. In some cases, like the projection on the plane the distinction between normal vectors as mentioned before is just not relevant.
answered 20 hours ago
weee
4308
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
It depends on what you consider two-sided. Let's think of your simple example of the surface of the unit sphere in $mathbb R^3$. We fix the norh pole $x=(0,0,1)$.
If you want to move from $x$ out of the plane, you can move either inside or outside the sphere. This is due to the fact, that there exist two possible normal vectors. The distinction of these normal vectors is important for example in Gauss' Thorem.
If you think of sides similar to sides of a cube i.e. surfaces that enclose your set, or boundaries, then a plane consists only of one boundary. If you consider again the north pose $x$ it does not lie on the inside or outside of the surface of the sphere, and the same holds for every point on any plane. In some cases, like the projection on the plane the distinction between normal vectors as mentioned before is just not relevant.
answered 20 hours ago
weee
4308
add a comment |
up vote
0
down vote
It depends on what you consider two-sided. Let's think of your simple example of the surface of the unit sphere in $mathbb R^3$. We fix the norh pole $x=(0,0,1)$.
If you want to move from $x$ out of the plane, you can move either inside or outside the sphere. This is due to the fact, that there exist two possible normal vectors. The distinction of these normal vectors is important for example in Gauss' Thorem.
If you think of sides similar to sides of a cube i.e. surfaces that enclose your set, or boundaries, then a plane consists only of one boundary. If you consider again the north pose $x$ it does not lie on the inside or outside of the surface of the sphere, and the same holds for every point on any plane. In some cases, like the projection on the plane the distinction between normal vectors as mentioned before is just not relevant.
answered 20 hours ago
weee
4308
add a comment |
up vote
0
down vote
up vote
0
down vote
It depends on what you consider two-sided. Let's think of your simple example of the surface of the unit sphere in $mathbb R^3$. We fix the norh pole $x=(0,0,1)$.
If you want to move from $x$ out of the plane, you can move either inside or outside the sphere. This is due to the fact, that there exist two possible normal vectors. The distinction of these normal vectors is important for example in Gauss' Thorem.
If you think of sides similar to sides of a cube i.e. surfaces that enclose your set, or boundaries, then a plane consists only of one boundary. If you consider again the north pose $x$ it does not lie on the inside or outside of the surface of the sphere, and the same holds for every point on any plane. In some cases, like the projection on the plane the distinction between normal vectors as mentioned before is just not relevant.
answered 20 hours ago
weee
4308
It depends on what you consider two-sided. Let's think of your simple example of the surface of the unit sphere in $mathbb R^3$. We fix the norh pole $x=(0,0,1)$.
If you want to move from $x$ out of the plane, you can move either inside or outside the sphere. This is due to the fact, that there exist two possible normal vectors. The distinction of these normal vectors is important for example in Gauss' Thorem.
If you think of sides similar to sides of a cube i.e. surfaces that enclose your set, or boundaries, then a plane consists only of one boundary. If you consider again the north pose $x$ it does not lie on the inside or outside of the surface of the sphere, and the same holds for every point on any plane. In some cases, like the projection on the plane the distinction between normal vectors as mentioned before is just not relevant.
answered 20 hours ago
weee
4308
answered 20 hours ago
weee
4308
answered 20 hours ago
weee
4308
answered 20 hours ago
weee
4308
4308
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there is no sides of planes in the standard geometry. A projection into a plane is part of the plane, so it have "two sides" or "one side", whatever you want to consider
– Masacroso
22 hours ago
A plane has no sides, conventionally. The space around the plane, however, does. And there are applications where surfaces are considered to have sides. See, for instance, this video about turning spheres inside-out. Here the sphere surface does have two sides (a yellow and a purple), and there is nothing stopping you from making a plane from the same "material".
– Arthur
22 hours ago
Thanks - I read: "A plane is a flat, two-dimensional surface." As a 'muggle', I find it difficult to comprehend the notion of a surface that has no sides. It's why I asked for some sort of illustration rather than merely a statement of fact.
– Konchog
22 hours ago
@Konchog I think you would need to define/describe what you mean by 'two-sided'. 'Visually' it makes sense if I think of a sheet of paper for example. However even thinking of the projection you describe, it does not 'need' two sides: A vector/point in the plane is just a vector/point in the plane.
– AnyAD
21 hours ago
A projection of some $Ssubset Bbb R^3$ to a plane $Psubset Bbb R^3$ maps each point of $S$ to a point of $P.$ And a point in $P$ does not have a "side". It does not belong to the "top" or "bottom" of $P.$ In everyday physical reality we do not encounter infinitely thin objects with zero volume.
– DanielWainfleet
20 hours ago