Why are map projections of the Earth not charts?
$begingroup$
The Earth is a classic example of a 2D manifold. Looks Euclidean to us, but is most definitely curved.
I am self teaching some differential geometry and I don't quite understand the difference between two things.
So I understand that a chart is a mapping between a subset of the manifold and Eulclidean space, but what I don't understand is how this relates to maps of the Earth.
I have read that it is impossible to represent the surface of the Earth/sphere with just a single chart. However, there are definitely maps of the earth that encompass every point. I have seen one that is centered around the north pole, for example, that includes the whole surface - except the south pole is definitely misrepresented somewhat, as it is wrapped around the edges.
So what is it about map projections that means they are not considered charts?
One presumes some level of maths is needed to perform a map projection, are there specific rules for what makes a chart a chart? And presumably these rules are broken for map projections?
differential-geometry manifolds
asked Dec 11 '18 at 11:59
bidbybidby
977
$endgroup$
add a comment |
$begingroup$
The Earth is a classic example of a 2D manifold. Looks Euclidean to us, but is most definitely curved.
I am self teaching some differential geometry and I don't quite understand the difference between two things.
So I understand that a chart is a mapping between a subset of the manifold and Eulclidean space, but what I don't understand is how this relates to maps of the Earth.
I have read that it is impossible to represent the surface of the Earth/sphere with just a single chart. However, there are definitely maps of the earth that encompass every point. I have seen one that is centered around the north pole, for example, that includes the whole surface - except the south pole is definitely misrepresented somewhat, as it is wrapped around the edges.
So what is it about map projections that means they are not considered charts?
One presumes some level of maths is needed to perform a map projection, are there specific rules for what makes a chart a chart? And presumably these rules are broken for map projections?
differential-geometry manifolds
asked Dec 11 '18 at 11:59
bidbybidby
977
$endgroup$
$begingroup$
What comes after I have also read about how every map?
$endgroup$
– timtfj
Dec 11 '18 at 12:10
$begingroup$
How can it encompass every point but also miss the south pole? That’s contradictory.
$endgroup$
– Randall
Dec 11 '18 at 13:54
$begingroup$
as David K pointed out, the south pole is represented by many points round the edge, and that map projection is therefore not a one-to-one function thus explaining the origin of my misunderstanding.
$endgroup$
– bidby
Dec 11 '18 at 13:56
add a comment |
$begingroup$
The Earth is a classic example of a 2D manifold. Looks Euclidean to us, but is most definitely curved.
I am self teaching some differential geometry and I don't quite understand the difference between two things.
So I understand that a chart is a mapping between a subset of the manifold and Eulclidean space, but what I don't understand is how this relates to maps of the Earth.
I have read that it is impossible to represent the surface of the Earth/sphere with just a single chart. However, there are definitely maps of the earth that encompass every point. I have seen one that is centered around the north pole, for example, that includes the whole surface - except the south pole is definitely misrepresented somewhat, as it is wrapped around the edges.
So what is it about map projections that means they are not considered charts?
One presumes some level of maths is needed to perform a map projection, are there specific rules for what makes a chart a chart? And presumably these rules are broken for map projections?
differential-geometry manifolds
asked Dec 11 '18 at 11:59
bidbybidby
977
$endgroup$
The Earth is a classic example of a 2D manifold. Looks Euclidean to us, but is most definitely curved.
I am self teaching some differential geometry and I don't quite understand the difference between two things.
So I understand that a chart is a mapping between a subset of the manifold and Eulclidean space, but what I don't understand is how this relates to maps of the Earth.
I have read that it is impossible to represent the surface of the Earth/sphere with just a single chart. However, there are definitely maps of the earth that encompass every point. I have seen one that is centered around the north pole, for example, that includes the whole surface - except the south pole is definitely misrepresented somewhat, as it is wrapped around the edges.
So what is it about map projections that means they are not considered charts?
One presumes some level of maths is needed to perform a map projection, are there specific rules for what makes a chart a chart? And presumably these rules are broken for map projections?
differential-geometry manifolds
differential-geometry manifolds
asked Dec 11 '18 at 11:59
bidbybidby
977
asked Dec 11 '18 at 11:59
bidbybidby
977
edited Dec 11 '18 at 13:07
bidby
asked Dec 11 '18 at 11:59
bidbybidby
977
asked Dec 11 '18 at 11:59
bidbybidby
977
asked Dec 11 '18 at 11:59
bidbybidby
977
977
$begingroup$
What comes after I have also read about how every map?
$endgroup$
– timtfj
Dec 11 '18 at 12:10
$begingroup$
How can it encompass every point but also miss the south pole? That’s contradictory.
$endgroup$
– Randall
Dec 11 '18 at 13:54
$begingroup$
as David K pointed out, the south pole is represented by many points round the edge, and that map projection is therefore not a one-to-one function thus explaining the origin of my misunderstanding.
$endgroup$
– bidby
Dec 11 '18 at 13:56
add a comment |
$begingroup$
What comes after I have also read about how every map?
$endgroup$
– timtfj
Dec 11 '18 at 12:10
$begingroup$
How can it encompass every point but also miss the south pole? That’s contradictory.
$endgroup$
– Randall
Dec 11 '18 at 13:54
$begingroup$
as David K pointed out, the south pole is represented by many points round the edge, and that map projection is therefore not a one-to-one function thus explaining the origin of my misunderstanding.
$endgroup$
– bidby
Dec 11 '18 at 13:56
$begingroup$
What comes after I have also read about how every map?
$endgroup$
– timtfj
Dec 11 '18 at 12:10
$begingroup$
What comes after I have also read about how every map?
$endgroup$
– timtfj
Dec 11 '18 at 12:10
$begingroup$
How can it encompass every point but also miss the south pole? That’s contradictory.
$endgroup$
– Randall
Dec 11 '18 at 13:54
$begingroup$
How can it encompass every point but also miss the south pole? That’s contradictory.
$endgroup$
– Randall
Dec 11 '18 at 13:54
$begingroup$
as David K pointed out, the south pole is represented by many points round the edge, and that map projection is therefore not a one-to-one function thus explaining the origin of my misunderstanding.
$endgroup$
– bidby
Dec 11 '18 at 13:56
$begingroup$
as David K pointed out, the south pole is represented by many points round the edge, and that map projection is therefore not a one-to-one function thus explaining the origin of my misunderstanding.
$endgroup$
– bidby
Dec 11 '18 at 13:56
add a comment |
1 Answer
1
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oldest
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$begingroup$
A mapping $U to mathbb R^n$ from an open subset $U$ of your manifold
is not a chart if one point of $U$ is "wrapped around the edges."
You may have answered your own question with that single sentence.
The problem isn't that it's impossible to make a map projection covering every point on Earth--the problem is that every such projection represents at least one point too many times. It gives a one-to-many correspondence that is not a function.
Alternatively, if you erase the redundant copies of each such point and keep just one, you still don't have a continuous mapping at that point, so you still don't have a chart in the mathematical sense.
If you remove one point from a sphere $S$ to get an open set $S_1,$
a stereographic projection gives a nice conformal mapping $S_1 to mathbb R^2.$
In mathematics, however, a sphere with one point removed is not a sphere.
It's a sphere with one point removed.
answered Dec 11 '18 at 12:55
David KDavid K
55.4k344120
$endgroup$
add a comment |
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1 Answer
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active
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1 Answer
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$begingroup$
A mapping $U to mathbb R^n$ from an open subset $U$ of your manifold
is not a chart if one point of $U$ is "wrapped around the edges."
You may have answered your own question with that single sentence.
The problem isn't that it's impossible to make a map projection covering every point on Earth--the problem is that every such projection represents at least one point too many times. It gives a one-to-many correspondence that is not a function.
Alternatively, if you erase the redundant copies of each such point and keep just one, you still don't have a continuous mapping at that point, so you still don't have a chart in the mathematical sense.
If you remove one point from a sphere $S$ to get an open set $S_1,$
a stereographic projection gives a nice conformal mapping $S_1 to mathbb R^2.$
In mathematics, however, a sphere with one point removed is not a sphere.
It's a sphere with one point removed.
answered Dec 11 '18 at 12:55
David KDavid K
55.4k344120
$endgroup$
add a comment |
$begingroup$
A mapping $U to mathbb R^n$ from an open subset $U$ of your manifold
is not a chart if one point of $U$ is "wrapped around the edges."
You may have answered your own question with that single sentence.
The problem isn't that it's impossible to make a map projection covering every point on Earth--the problem is that every such projection represents at least one point too many times. It gives a one-to-many correspondence that is not a function.
Alternatively, if you erase the redundant copies of each such point and keep just one, you still don't have a continuous mapping at that point, so you still don't have a chart in the mathematical sense.
If you remove one point from a sphere $S$ to get an open set $S_1,$
a stereographic projection gives a nice conformal mapping $S_1 to mathbb R^2.$
In mathematics, however, a sphere with one point removed is not a sphere.
It's a sphere with one point removed.
answered Dec 11 '18 at 12:55
David KDavid K
55.4k344120
$endgroup$
add a comment |
$begingroup$
A mapping $U to mathbb R^n$ from an open subset $U$ of your manifold
is not a chart if one point of $U$ is "wrapped around the edges."
You may have answered your own question with that single sentence.
The problem isn't that it's impossible to make a map projection covering every point on Earth--the problem is that every such projection represents at least one point too many times. It gives a one-to-many correspondence that is not a function.
Alternatively, if you erase the redundant copies of each such point and keep just one, you still don't have a continuous mapping at that point, so you still don't have a chart in the mathematical sense.
If you remove one point from a sphere $S$ to get an open set $S_1,$
a stereographic projection gives a nice conformal mapping $S_1 to mathbb R^2.$
In mathematics, however, a sphere with one point removed is not a sphere.
It's a sphere with one point removed.
answered Dec 11 '18 at 12:55
David KDavid K
55.4k344120
$endgroup$
A mapping $U to mathbb R^n$ from an open subset $U$ of your manifold
is not a chart if one point of $U$ is "wrapped around the edges."
You may have answered your own question with that single sentence.
The problem isn't that it's impossible to make a map projection covering every point on Earth--the problem is that every such projection represents at least one point too many times. It gives a one-to-many correspondence that is not a function.
Alternatively, if you erase the redundant copies of each such point and keep just one, you still don't have a continuous mapping at that point, so you still don't have a chart in the mathematical sense.
If you remove one point from a sphere $S$ to get an open set $S_1,$
a stereographic projection gives a nice conformal mapping $S_1 to mathbb R^2.$
In mathematics, however, a sphere with one point removed is not a sphere.
It's a sphere with one point removed.
answered Dec 11 '18 at 12:55
David KDavid K
55.4k344120
edited Dec 11 '18 at 13:01
answered Dec 11 '18 at 12:55
David KDavid K
55.4k344120
answered Dec 11 '18 at 12:55
David KDavid K
55.4k344120
answered Dec 11 '18 at 12:55
David KDavid K
55.4k344120
55.4k344120
add a comment |
add a comment |
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$begingroup$
What comes after I have also read about how every map?
$endgroup$
– timtfj
Dec 11 '18 at 12:10
$begingroup$
How can it encompass every point but also miss the south pole? That’s contradictory.
$endgroup$
– Randall
Dec 11 '18 at 13:54
$begingroup$
as David K pointed out, the south pole is represented by many points round the edge, and that map projection is therefore not a one-to-one function thus explaining the origin of my misunderstanding.
$endgroup$
– bidby
Dec 11 '18 at 13:56