When does it make sense to find a point between two points?
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I have this need to be able to express when a point is "between" two other points. One great example is the binary average operation $Avg:R times R rightarrow R$ that takes two real numbers and produces the average of the two (like a mid-point).
Here is a draft of how I would go about defining when a point is between two.
Let M be a metric space with three points a,b and c.
A point b is said to be between points a and c iff:
$d(a,b) + d(b,c) = d(a,c)$
Is a metric space enough to determine when a point is between two other points through the triangle equality, or will I need some other machinery? Is this the best approach?
geometry metric-spaces average
asked Dec 7 '18 at 0:03
user512716user512716
816
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add a comment |
$begingroup$
I have this need to be able to express when a point is "between" two other points. One great example is the binary average operation $Avg:R times R rightarrow R$ that takes two real numbers and produces the average of the two (like a mid-point).
Here is a draft of how I would go about defining when a point is between two.
Let M be a metric space with three points a,b and c.
A point b is said to be between points a and c iff:
$d(a,b) + d(b,c) = d(a,c)$
Is a metric space enough to determine when a point is between two other points through the triangle equality, or will I need some other machinery? Is this the best approach?
geometry metric-spaces average
asked Dec 7 '18 at 0:03
user512716user512716
816
$endgroup$
$begingroup$
Do you want the point $b$ to be unique? In Euclidean space your definition is satisfied by all points on the line segment joining $a$ and $c$.
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– Rahul
Dec 7 '18 at 0:35
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@Rahul No. Since I want to formalize the notion of a line segment in a metric space.
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– user512716
Dec 7 '18 at 0:38
1
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OK. I was only concerned because your motivating example was a midpoint instead of a line segment. An interesting fact is that under the Manhattan metric a "line segment" is actually a rectangle.
$endgroup$
– Rahul
Dec 7 '18 at 1:31
add a comment |
$begingroup$
I have this need to be able to express when a point is "between" two other points. One great example is the binary average operation $Avg:R times R rightarrow R$ that takes two real numbers and produces the average of the two (like a mid-point).
Here is a draft of how I would go about defining when a point is between two.
Let M be a metric space with three points a,b and c.
A point b is said to be between points a and c iff:
$d(a,b) + d(b,c) = d(a,c)$
Is a metric space enough to determine when a point is between two other points through the triangle equality, or will I need some other machinery? Is this the best approach?
geometry metric-spaces average
asked Dec 7 '18 at 0:03
user512716user512716
816
$endgroup$
I have this need to be able to express when a point is "between" two other points. One great example is the binary average operation $Avg:R times R rightarrow R$ that takes two real numbers and produces the average of the two (like a mid-point).
Here is a draft of how I would go about defining when a point is between two.
Let M be a metric space with three points a,b and c.
A point b is said to be between points a and c iff:
$d(a,b) + d(b,c) = d(a,c)$
Is a metric space enough to determine when a point is between two other points through the triangle equality, or will I need some other machinery? Is this the best approach?
geometry metric-spaces average
geometry metric-spaces average
asked Dec 7 '18 at 0:03
user512716user512716
816
asked Dec 7 '18 at 0:03
user512716user512716
816
asked Dec 7 '18 at 0:03
user512716user512716
816
asked Dec 7 '18 at 0:03
user512716user512716
816
asked Dec 7 '18 at 0:03
user512716user512716
816
816
$begingroup$
Do you want the point $b$ to be unique? In Euclidean space your definition is satisfied by all points on the line segment joining $a$ and $c$.
$endgroup$
– Rahul
Dec 7 '18 at 0:35
$begingroup$
@Rahul No. Since I want to formalize the notion of a line segment in a metric space.
$endgroup$
– user512716
Dec 7 '18 at 0:38
1
$begingroup$
OK. I was only concerned because your motivating example was a midpoint instead of a line segment. An interesting fact is that under the Manhattan metric a "line segment" is actually a rectangle.
$endgroup$
– Rahul
Dec 7 '18 at 1:31
add a comment |
$begingroup$
Do you want the point $b$ to be unique? In Euclidean space your definition is satisfied by all points on the line segment joining $a$ and $c$.
$endgroup$
– Rahul
Dec 7 '18 at 0:35
$begingroup$
@Rahul No. Since I want to formalize the notion of a line segment in a metric space.
$endgroup$
– user512716
Dec 7 '18 at 0:38
1
$begingroup$
OK. I was only concerned because your motivating example was a midpoint instead of a line segment. An interesting fact is that under the Manhattan metric a "line segment" is actually a rectangle.
$endgroup$
– Rahul
Dec 7 '18 at 1:31
$begingroup$
Do you want the point $b$ to be unique? In Euclidean space your definition is satisfied by all points on the line segment joining $a$ and $c$.
$endgroup$
– Rahul
Dec 7 '18 at 0:35
$begingroup$
Do you want the point $b$ to be unique? In Euclidean space your definition is satisfied by all points on the line segment joining $a$ and $c$.
$endgroup$
– Rahul
Dec 7 '18 at 0:35
$begingroup$
@Rahul No. Since I want to formalize the notion of a line segment in a metric space.
$endgroup$
– user512716
Dec 7 '18 at 0:38
$begingroup$
@Rahul No. Since I want to formalize the notion of a line segment in a metric space.
$endgroup$
– user512716
Dec 7 '18 at 0:38
1
1
$begingroup$
OK. I was only concerned because your motivating example was a midpoint instead of a line segment. An interesting fact is that under the Manhattan metric a "line segment" is actually a rectangle.
$endgroup$
– Rahul
Dec 7 '18 at 1:31
$begingroup$
OK. I was only concerned because your motivating example was a midpoint instead of a line segment. An interesting fact is that under the Manhattan metric a "line segment" is actually a rectangle.
$endgroup$
– Rahul
Dec 7 '18 at 1:31
add a comment |
1 Answer
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There is nothing inherently wrong with this definition. Indeed, so long as you are fine with the possibility that there are no points between anything (the discrete metric) then you are good to go. Depending on the metric you may find it hard to compute these "between" points, but that is perhaps a question best saved for the specific metric space you have in mind.
answered Dec 7 '18 at 0:27
RandomMathGuyRandomMathGuy
1192
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1 Answer
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$begingroup$
There is nothing inherently wrong with this definition. Indeed, so long as you are fine with the possibility that there are no points between anything (the discrete metric) then you are good to go. Depending on the metric you may find it hard to compute these "between" points, but that is perhaps a question best saved for the specific metric space you have in mind.
answered Dec 7 '18 at 0:27
RandomMathGuyRandomMathGuy
1192
$endgroup$
add a comment |
$begingroup$
There is nothing inherently wrong with this definition. Indeed, so long as you are fine with the possibility that there are no points between anything (the discrete metric) then you are good to go. Depending on the metric you may find it hard to compute these "between" points, but that is perhaps a question best saved for the specific metric space you have in mind.
answered Dec 7 '18 at 0:27
RandomMathGuyRandomMathGuy
1192
$endgroup$
add a comment |
$begingroup$
There is nothing inherently wrong with this definition. Indeed, so long as you are fine with the possibility that there are no points between anything (the discrete metric) then you are good to go. Depending on the metric you may find it hard to compute these "between" points, but that is perhaps a question best saved for the specific metric space you have in mind.
answered Dec 7 '18 at 0:27
RandomMathGuyRandomMathGuy
1192
$endgroup$
There is nothing inherently wrong with this definition. Indeed, so long as you are fine with the possibility that there are no points between anything (the discrete metric) then you are good to go. Depending on the metric you may find it hard to compute these "between" points, but that is perhaps a question best saved for the specific metric space you have in mind.
answered Dec 7 '18 at 0:27
RandomMathGuyRandomMathGuy
1192
answered Dec 7 '18 at 0:27
RandomMathGuyRandomMathGuy
1192
answered Dec 7 '18 at 0:27
RandomMathGuyRandomMathGuy
1192
answered Dec 7 '18 at 0:27
RandomMathGuyRandomMathGuy
1192
1192
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$begingroup$
Do you want the point $b$ to be unique? In Euclidean space your definition is satisfied by all points on the line segment joining $a$ and $c$.
$endgroup$
– Rahul
Dec 7 '18 at 0:35
$begingroup$
@Rahul No. Since I want to formalize the notion of a line segment in a metric space.
$endgroup$
– user512716
Dec 7 '18 at 0:38
1
$begingroup$
OK. I was only concerned because your motivating example was a midpoint instead of a line segment. An interesting fact is that under the Manhattan metric a "line segment" is actually a rectangle.
$endgroup$
– Rahul
Dec 7 '18 at 1:31