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What makes a line “straight”?
The Next CEO of Stack OverflowDefinition of Geodesic - Distance between two points on the same latitudeGeodesics (1): Spaces with more than two geodesics between two pointsGeodesics (2): Is the real projective plane intended to make shortest paths unique?Is there a surface in Euclidean space that admits elliptic geometry?Why is a straight line the shortest distance between two points?How to construct a line with a given equal distance from 3 Points in 3 Dimensions?Mapping sphere surface to a vector space such that distances are preserved?Is the shortest distance between two points in Euclidean $n$-space always a straight line?How to generalize parallelograms to non-Euclidean spaces?Distance between a line segment and a point equation
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In Euclidean space, there can be several definitions that makes a straght line:
- Line of shortest distance between two points
- Line that is linear, i.e with the points satisfying a linear equation
- Line of zero curvature everywhere
- Line with constant curvature that is equal from both sides
The Wikipedia article states:
In geometry, it is frequently the case that the concept of line is taken as a primitive.
When one enters non-Euclidean geometry, are the above definitions consistent with each other? Definition 1, for example, becomes a geodesic.
geometry definition
asked Oct 11 '14 at 11:44
user158163user158163
685
$endgroup$
|
show 5 more comments
$begingroup$
In Euclidean space, there can be several definitions that makes a straght line:
- Line of shortest distance between two points
- Line that is linear, i.e with the points satisfying a linear equation
- Line of zero curvature everywhere
- Line with constant curvature that is equal from both sides
The Wikipedia article states:
In geometry, it is frequently the case that the concept of line is taken as a primitive.
When one enters non-Euclidean geometry, are the above definitions consistent with each other? Definition 1, for example, becomes a geodesic.
geometry definition
asked Oct 11 '14 at 11:44
user158163user158163
685
$endgroup$
$begingroup$
in differential geometry def. 1 is geodesic but def.2,def.3 define the straight line.
$endgroup$
– Haha
Oct 11 '14 at 12:05
1
$begingroup$
"linear equation" presupposes coordinates, which you may not necessarily have in nonEuclidean geometry.
$endgroup$
– Gerry Myerson
Oct 11 '14 at 12:14
$begingroup$
Similarly, curvature entails a notion of limits, which doesn't make sense in, say, finite projective geometries. So 3 and 4 don't generalize terrible well either. And condition 1 requires a notion of distance, which also doesn't apply in finite projective geometries, like the 7-point projective plane.
$endgroup$
– John Hughes
Oct 11 '14 at 12:53
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So, there does not exist a single definition that is applicable to everywhere? (Or, almost everywhere?)
$endgroup$
– user158163
Oct 11 '14 at 13:48
$begingroup$
Firstly, the quote from Wikipedia is for "line" only. "Taken as a primitive" means that there is no attempt to define it. Adding "straight" starts a whole new ball game. To differentiate types of lines from each other by shape, you need something like coordinates and distance. Otherwise, you can only discuss what is and what isn't a line.
$endgroup$
– rschwieb
Oct 11 '14 at 13:58
|
show 5 more comments
$begingroup$
In Euclidean space, there can be several definitions that makes a straght line:
- Line of shortest distance between two points
- Line that is linear, i.e with the points satisfying a linear equation
- Line of zero curvature everywhere
- Line with constant curvature that is equal from both sides
The Wikipedia article states:
In geometry, it is frequently the case that the concept of line is taken as a primitive.
When one enters non-Euclidean geometry, are the above definitions consistent with each other? Definition 1, for example, becomes a geodesic.
geometry definition
asked Oct 11 '14 at 11:44
user158163user158163
685
$endgroup$
In Euclidean space, there can be several definitions that makes a straght line:
- Line of shortest distance between two points
- Line that is linear, i.e with the points satisfying a linear equation
- Line of zero curvature everywhere
- Line with constant curvature that is equal from both sides
The Wikipedia article states:
In geometry, it is frequently the case that the concept of line is taken as a primitive.
When one enters non-Euclidean geometry, are the above definitions consistent with each other? Definition 1, for example, becomes a geodesic.
geometry definition
geometry definition
asked Oct 11 '14 at 11:44
user158163user158163
685
asked Oct 11 '14 at 11:44
user158163user158163
685
edited Oct 11 '14 at 11:49
user158163
asked Oct 11 '14 at 11:44
user158163user158163
685
asked Oct 11 '14 at 11:44
user158163user158163
685
asked Oct 11 '14 at 11:44
user158163user158163
685
685
$begingroup$
in differential geometry def. 1 is geodesic but def.2,def.3 define the straight line.
$endgroup$
– Haha
Oct 11 '14 at 12:05
1
$begingroup$
"linear equation" presupposes coordinates, which you may not necessarily have in nonEuclidean geometry.
$endgroup$
– Gerry Myerson
Oct 11 '14 at 12:14
$begingroup$
Similarly, curvature entails a notion of limits, which doesn't make sense in, say, finite projective geometries. So 3 and 4 don't generalize terrible well either. And condition 1 requires a notion of distance, which also doesn't apply in finite projective geometries, like the 7-point projective plane.
$endgroup$
– John Hughes
Oct 11 '14 at 12:53
$begingroup$
So, there does not exist a single definition that is applicable to everywhere? (Or, almost everywhere?)
$endgroup$
– user158163
Oct 11 '14 at 13:48
$begingroup$
Firstly, the quote from Wikipedia is for "line" only. "Taken as a primitive" means that there is no attempt to define it. Adding "straight" starts a whole new ball game. To differentiate types of lines from each other by shape, you need something like coordinates and distance. Otherwise, you can only discuss what is and what isn't a line.
$endgroup$
– rschwieb
Oct 11 '14 at 13:58
|
show 5 more comments
$begingroup$
in differential geometry def. 1 is geodesic but def.2,def.3 define the straight line.
$endgroup$
– Haha
Oct 11 '14 at 12:05
1
$begingroup$
"linear equation" presupposes coordinates, which you may not necessarily have in nonEuclidean geometry.
$endgroup$
– Gerry Myerson
Oct 11 '14 at 12:14
$begingroup$
Similarly, curvature entails a notion of limits, which doesn't make sense in, say, finite projective geometries. So 3 and 4 don't generalize terrible well either. And condition 1 requires a notion of distance, which also doesn't apply in finite projective geometries, like the 7-point projective plane.
$endgroup$
– John Hughes
Oct 11 '14 at 12:53
$begingroup$
So, there does not exist a single definition that is applicable to everywhere? (Or, almost everywhere?)
$endgroup$
– user158163
Oct 11 '14 at 13:48
$begingroup$
Firstly, the quote from Wikipedia is for "line" only. "Taken as a primitive" means that there is no attempt to define it. Adding "straight" starts a whole new ball game. To differentiate types of lines from each other by shape, you need something like coordinates and distance. Otherwise, you can only discuss what is and what isn't a line.
$endgroup$
– rschwieb
Oct 11 '14 at 13:58
$begingroup$
in differential geometry def. 1 is geodesic but def.2,def.3 define the straight line.
$endgroup$
– Haha
Oct 11 '14 at 12:05
$begingroup$
in differential geometry def. 1 is geodesic but def.2,def.3 define the straight line.
$endgroup$
– Haha
Oct 11 '14 at 12:05
1
1
$begingroup$
"linear equation" presupposes coordinates, which you may not necessarily have in nonEuclidean geometry.
$endgroup$
– Gerry Myerson
Oct 11 '14 at 12:14
$begingroup$
"linear equation" presupposes coordinates, which you may not necessarily have in nonEuclidean geometry.
$endgroup$
– Gerry Myerson
Oct 11 '14 at 12:14
$begingroup$
Similarly, curvature entails a notion of limits, which doesn't make sense in, say, finite projective geometries. So 3 and 4 don't generalize terrible well either. And condition 1 requires a notion of distance, which also doesn't apply in finite projective geometries, like the 7-point projective plane.
$endgroup$
– John Hughes
Oct 11 '14 at 12:53
$begingroup$
Similarly, curvature entails a notion of limits, which doesn't make sense in, say, finite projective geometries. So 3 and 4 don't generalize terrible well either. And condition 1 requires a notion of distance, which also doesn't apply in finite projective geometries, like the 7-point projective plane.
$endgroup$
– John Hughes
Oct 11 '14 at 12:53
$begingroup$
So, there does not exist a single definition that is applicable to everywhere? (Or, almost everywhere?)
$endgroup$
– user158163
Oct 11 '14 at 13:48
$begingroup$
So, there does not exist a single definition that is applicable to everywhere? (Or, almost everywhere?)
$endgroup$
– user158163
Oct 11 '14 at 13:48
$begingroup$
Firstly, the quote from Wikipedia is for "line" only. "Taken as a primitive" means that there is no attempt to define it. Adding "straight" starts a whole new ball game. To differentiate types of lines from each other by shape, you need something like coordinates and distance. Otherwise, you can only discuss what is and what isn't a line.
$endgroup$
– rschwieb
Oct 11 '14 at 13:58
$begingroup$
Firstly, the quote from Wikipedia is for "line" only. "Taken as a primitive" means that there is no attempt to define it. Adding "straight" starts a whole new ball game. To differentiate types of lines from each other by shape, you need something like coordinates and distance. Otherwise, you can only discuss what is and what isn't a line.
$endgroup$
– rschwieb
Oct 11 '14 at 13:58
|
show 5 more comments
1 Answer
1
active
oldest
votes
$begingroup$
None of these definitions as satisfactory as they rely on other concepts, such as distance or curvature, even less defined.
answered Feb 10 at 14:08
Yves DaoustYves Daoust
131k676229
$endgroup$
add a comment |
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$begingroup$
None of these definitions as satisfactory as they rely on other concepts, such as distance or curvature, even less defined.
answered Feb 10 at 14:08
Yves DaoustYves Daoust
131k676229
$endgroup$
add a comment |
$begingroup$
None of these definitions as satisfactory as they rely on other concepts, such as distance or curvature, even less defined.
answered Feb 10 at 14:08
Yves DaoustYves Daoust
131k676229
$endgroup$
add a comment |
$begingroup$
None of these definitions as satisfactory as they rely on other concepts, such as distance or curvature, even less defined.
answered Feb 10 at 14:08
Yves DaoustYves Daoust
131k676229
$endgroup$
None of these definitions as satisfactory as they rely on other concepts, such as distance or curvature, even less defined.
answered Feb 10 at 14:08
Yves DaoustYves Daoust
131k676229
answered Feb 10 at 14:08
Yves DaoustYves Daoust
131k676229
answered Feb 10 at 14:08
Yves DaoustYves Daoust
131k676229
answered Feb 10 at 14:08
Yves DaoustYves Daoust
131k676229
131k676229
add a comment |
add a comment |
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$begingroup$
in differential geometry def. 1 is geodesic but def.2,def.3 define the straight line.
$endgroup$
– Haha
Oct 11 '14 at 12:05
1
$begingroup$
"linear equation" presupposes coordinates, which you may not necessarily have in nonEuclidean geometry.
$endgroup$
– Gerry Myerson
Oct 11 '14 at 12:14
$begingroup$
Similarly, curvature entails a notion of limits, which doesn't make sense in, say, finite projective geometries. So 3 and 4 don't generalize terrible well either. And condition 1 requires a notion of distance, which also doesn't apply in finite projective geometries, like the 7-point projective plane.
$endgroup$
– John Hughes
Oct 11 '14 at 12:53
$begingroup$
So, there does not exist a single definition that is applicable to everywhere? (Or, almost everywhere?)
$endgroup$
– user158163
Oct 11 '14 at 13:48
$begingroup$
Firstly, the quote from Wikipedia is for "line" only. "Taken as a primitive" means that there is no attempt to define it. Adding "straight" starts a whole new ball game. To differentiate types of lines from each other by shape, you need something like coordinates and distance. Otherwise, you can only discuss what is and what isn't a line.
$endgroup$
– rschwieb
Oct 11 '14 at 13:58