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What does it mean for something to be a model of hyperbolic space?
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In the book "Non-Euclidean Geometry and Curvature" by James W. Cannon, the author uses the term "analytic models of hyperbolic space." (p. 19) Some examples are the Klein model and the Hyperboloid model, which are mentioned on Wikipedia as well. However, he does not explain what such a model is. Why do they qualify as models of hyperbolic space, and how can both be valid?
hyperbolic-geometry curvature
asked Jul 17 at 9:56
Tensor McTensorstein
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up vote
8
down vote
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In the book "Non-Euclidean Geometry and Curvature" by James W. Cannon, the author uses the term "analytic models of hyperbolic space." (p. 19) Some examples are the Klein model and the Hyperboloid model, which are mentioned on Wikipedia as well. However, he does not explain what such a model is. Why do they qualify as models of hyperbolic space, and how can both be valid?
hyperbolic-geometry curvature
asked Jul 17 at 9:56
Tensor McTensorstein
627
add a comment |Â
up vote
8
down vote
favorite
up vote
8
down vote
favorite
In the book "Non-Euclidean Geometry and Curvature" by James W. Cannon, the author uses the term "analytic models of hyperbolic space." (p. 19) Some examples are the Klein model and the Hyperboloid model, which are mentioned on Wikipedia as well. However, he does not explain what such a model is. Why do they qualify as models of hyperbolic space, and how can both be valid?
hyperbolic-geometry curvature
asked Jul 17 at 9:56
Tensor McTensorstein
627
In the book "Non-Euclidean Geometry and Curvature" by James W. Cannon, the author uses the term "analytic models of hyperbolic space." (p. 19) Some examples are the Klein model and the Hyperboloid model, which are mentioned on Wikipedia as well. However, he does not explain what such a model is. Why do they qualify as models of hyperbolic space, and how can both be valid?
hyperbolic-geometry curvature
asked Jul 17 at 9:56
Tensor McTensorstein
627
asked Jul 17 at 9:56
Tensor McTensorstein
627
asked Jul 17 at 9:56
Tensor McTensorstein
627
asked Jul 17 at 9:56
Tensor McTensorstein
627
627
add a comment |Â
add a comment |Â
4 Answers
4
active
oldest
votes
up vote
6
down vote
accepted
You are right - a model of hyperbolic space is a mathematical structure in which we define "points" and "lines" so that the modified Euclidean postulates (with the parallel postulate replaced by its hyperbolic equivalent) are true.
The 3 most common models of 2D hyperbolic space are the hyperboloid model, the Klein model and the Poincare model. In each model "points" are still normal geometric points, but "lines" are defined as geodesics in a non-Euclidean metric.
In the hyperboloid model the "lines" all lie on one sheet of a hyperboloid. In the Klein model "lines" are Euclidean lines on a plane but distance is no-Euclidean, so points at infinity lie on the circumference of a circle. In the Poincare model "lines" are arcs of circles on a plane; again, points at infinity lie on the circumference of a circle.
If you embed each of these 2D models in 3D Euclidean space then they can be related by very interesting and beautiful projections.
answered Jul 17 at 11:44
gandalf61
5,689522
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A model of a theory is a concrete, constructed example where the theory is applicable because the axioms are verified.
For example $mathbbZ$, $mathbbC$ and $S_6$ are all groups, meaning they are models for the group theory. They satisfy the group axioms and and you can apply group theory results to them. They are also very different from each other.
answered Jul 17 at 10:05
Rad80
1887
1
Alright. Does that, in this case, mean axioms like the modified version of Euclid's five postulates?
– Tensor McTensorstein
Jul 17 at 10:27
1
I would say so. However I have to say that I am not very familiar with hyperbolic space, much less with the approach used in a particular book.
– Rad80
Jul 17 at 10:41
Indeed the hyperbolic plane is a model of Euclid's axioms with the parallel postulate replaced by the postulate that for every line $L$ and every point $p$ not on $L$ there is more than one line through $p$ that is disjoint from $L$.
– Lee Mosher
Jul 18 at 2:59
add a comment |Â
up vote
1
down vote
There is another notion of "model" which characterizes the hyperbolic plane and is expressed in the language of differential geometry, namely:
The hyperbolic plane is the unique simply connected, complete Riemannian manifold of dimension 2 and constant curvature $-1$.
The meaning of "uniqueness" in this statement is uniqueness up to isometry. More formally, if $mathbb H_1$ and $mathbb H_2$ are two simply connected, complete Riemannian manifolds of dimension 2 and constant curvature $-1$ then there exists an isometry $f : mathbb H_1 to mathbb H_2$.
This notion of uniqueness can be used to prove that all of the models mentioned in your question are isometric to each other, by verifying that the Klein model, the hyperboloid model, and the Poincare model are all complete, simply connected, Riemannian 2-manifolds of constant curvature $-1$.
answered Jul 18 at 3:04
Lee Mosher
45.7k33478
add a comment |Â
up vote
0
down vote
I would say the word "model" here is used for historical purposes. The history is that people have been wondering whether Euclid's fifth postulate can be proven from the other postulates. This has been shown to be false by constructing a "model" (in the sense of the model theory), i.e., a structure which satisfies the other postulates, but not the fifth postulate. And thus we have models of non-Euclidean geometry.
Comparing with the spherical geometry could be useful. We can view the surface of Earth as a 3D object, or we can make a flat map of Earth using many well-known projections, e.g., stereographic, gnomonic, Mercator. If we apply the stereographic projection to the Minkowski hyperboloid, we get the Poincaré model of the hyperbolic plane. Likewise, gnomonic projection = Klein model, Mercator projection = band model, orthogonal = Gans model. Thus, the word "projection" would be appropriate for some of the models (we have to be very careful though, as the Minkowski hyperboloid lives in the Minkowski space, not the usual $mathbbR^3$) but "model" is still more common. I have written more about this here.
answered Jul 20 at 10:36
Zeno Rogue
33116
add a comment |Â
4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
down vote
accepted
You are right - a model of hyperbolic space is a mathematical structure in which we define "points" and "lines" so that the modified Euclidean postulates (with the parallel postulate replaced by its hyperbolic equivalent) are true.
The 3 most common models of 2D hyperbolic space are the hyperboloid model, the Klein model and the Poincare model. In each model "points" are still normal geometric points, but "lines" are defined as geodesics in a non-Euclidean metric.
In the hyperboloid model the "lines" all lie on one sheet of a hyperboloid. In the Klein model "lines" are Euclidean lines on a plane but distance is no-Euclidean, so points at infinity lie on the circumference of a circle. In the Poincare model "lines" are arcs of circles on a plane; again, points at infinity lie on the circumference of a circle.
If you embed each of these 2D models in 3D Euclidean space then they can be related by very interesting and beautiful projections.
answered Jul 17 at 11:44
gandalf61
5,689522
add a comment |Â
up vote
6
down vote
accepted
You are right - a model of hyperbolic space is a mathematical structure in which we define "points" and "lines" so that the modified Euclidean postulates (with the parallel postulate replaced by its hyperbolic equivalent) are true.
The 3 most common models of 2D hyperbolic space are the hyperboloid model, the Klein model and the Poincare model. In each model "points" are still normal geometric points, but "lines" are defined as geodesics in a non-Euclidean metric.
In the hyperboloid model the "lines" all lie on one sheet of a hyperboloid. In the Klein model "lines" are Euclidean lines on a plane but distance is no-Euclidean, so points at infinity lie on the circumference of a circle. In the Poincare model "lines" are arcs of circles on a plane; again, points at infinity lie on the circumference of a circle.
If you embed each of these 2D models in 3D Euclidean space then they can be related by very interesting and beautiful projections.
answered Jul 17 at 11:44
gandalf61
5,689522
add a comment |Â
up vote
6
down vote
accepted
up vote
6
down vote
accepted
You are right - a model of hyperbolic space is a mathematical structure in which we define "points" and "lines" so that the modified Euclidean postulates (with the parallel postulate replaced by its hyperbolic equivalent) are true.
The 3 most common models of 2D hyperbolic space are the hyperboloid model, the Klein model and the Poincare model. In each model "points" are still normal geometric points, but "lines" are defined as geodesics in a non-Euclidean metric.
In the hyperboloid model the "lines" all lie on one sheet of a hyperboloid. In the Klein model "lines" are Euclidean lines on a plane but distance is no-Euclidean, so points at infinity lie on the circumference of a circle. In the Poincare model "lines" are arcs of circles on a plane; again, points at infinity lie on the circumference of a circle.
If you embed each of these 2D models in 3D Euclidean space then they can be related by very interesting and beautiful projections.
answered Jul 17 at 11:44
gandalf61
5,689522
You are right - a model of hyperbolic space is a mathematical structure in which we define "points" and "lines" so that the modified Euclidean postulates (with the parallel postulate replaced by its hyperbolic equivalent) are true.
The 3 most common models of 2D hyperbolic space are the hyperboloid model, the Klein model and the Poincare model. In each model "points" are still normal geometric points, but "lines" are defined as geodesics in a non-Euclidean metric.
In the hyperboloid model the "lines" all lie on one sheet of a hyperboloid. In the Klein model "lines" are Euclidean lines on a plane but distance is no-Euclidean, so points at infinity lie on the circumference of a circle. In the Poincare model "lines" are arcs of circles on a plane; again, points at infinity lie on the circumference of a circle.
If you embed each of these 2D models in 3D Euclidean space then they can be related by very interesting and beautiful projections.
answered Jul 17 at 11:44
gandalf61
5,689522
answered Jul 17 at 11:44
gandalf61
5,689522
answered Jul 17 at 11:44
gandalf61
5,689522
answered Jul 17 at 11:44
gandalf61
5,689522
5,689522
add a comment |Â
add a comment |Â
up vote
3
down vote
A model of a theory is a concrete, constructed example where the theory is applicable because the axioms are verified.
For example $mathbbZ$, $mathbbC$ and $S_6$ are all groups, meaning they are models for the group theory. They satisfy the group axioms and and you can apply group theory results to them. They are also very different from each other.
answered Jul 17 at 10:05
Rad80
1887
1
Alright. Does that, in this case, mean axioms like the modified version of Euclid's five postulates?
– Tensor McTensorstein
Jul 17 at 10:27
1
I would say so. However I have to say that I am not very familiar with hyperbolic space, much less with the approach used in a particular book.
– Rad80
Jul 17 at 10:41
Indeed the hyperbolic plane is a model of Euclid's axioms with the parallel postulate replaced by the postulate that for every line $L$ and every point $p$ not on $L$ there is more than one line through $p$ that is disjoint from $L$.
– Lee Mosher
Jul 18 at 2:59
add a comment |Â
up vote
3
down vote
A model of a theory is a concrete, constructed example where the theory is applicable because the axioms are verified.
For example $mathbbZ$, $mathbbC$ and $S_6$ are all groups, meaning they are models for the group theory. They satisfy the group axioms and and you can apply group theory results to them. They are also very different from each other.
answered Jul 17 at 10:05
Rad80
1887
1
Alright. Does that, in this case, mean axioms like the modified version of Euclid's five postulates?
– Tensor McTensorstein
Jul 17 at 10:27
1
I would say so. However I have to say that I am not very familiar with hyperbolic space, much less with the approach used in a particular book.
– Rad80
Jul 17 at 10:41
Indeed the hyperbolic plane is a model of Euclid's axioms with the parallel postulate replaced by the postulate that for every line $L$ and every point $p$ not on $L$ there is more than one line through $p$ that is disjoint from $L$.
– Lee Mosher
Jul 18 at 2:59
add a comment |Â
up vote
3
down vote
up vote
3
down vote
A model of a theory is a concrete, constructed example where the theory is applicable because the axioms are verified.
For example $mathbbZ$, $mathbbC$ and $S_6$ are all groups, meaning they are models for the group theory. They satisfy the group axioms and and you can apply group theory results to them. They are also very different from each other.
answered Jul 17 at 10:05
Rad80
1887
A model of a theory is a concrete, constructed example where the theory is applicable because the axioms are verified.
For example $mathbbZ$, $mathbbC$ and $S_6$ are all groups, meaning they are models for the group theory. They satisfy the group axioms and and you can apply group theory results to them. They are also very different from each other.
answered Jul 17 at 10:05
Rad80
1887
answered Jul 17 at 10:05
Rad80
1887
answered Jul 17 at 10:05
Rad80
1887
answered Jul 17 at 10:05
Rad80
1887
1887
1
Alright. Does that, in this case, mean axioms like the modified version of Euclid's five postulates?
– Tensor McTensorstein
Jul 17 at 10:27
1
I would say so. However I have to say that I am not very familiar with hyperbolic space, much less with the approach used in a particular book.
– Rad80
Jul 17 at 10:41
Indeed the hyperbolic plane is a model of Euclid's axioms with the parallel postulate replaced by the postulate that for every line $L$ and every point $p$ not on $L$ there is more than one line through $p$ that is disjoint from $L$.
– Lee Mosher
Jul 18 at 2:59
add a comment |Â
1
Alright. Does that, in this case, mean axioms like the modified version of Euclid's five postulates?
– Tensor McTensorstein
Jul 17 at 10:27
1
I would say so. However I have to say that I am not very familiar with hyperbolic space, much less with the approach used in a particular book.
– Rad80
Jul 17 at 10:41
Indeed the hyperbolic plane is a model of Euclid's axioms with the parallel postulate replaced by the postulate that for every line $L$ and every point $p$ not on $L$ there is more than one line through $p$ that is disjoint from $L$.
– Lee Mosher
Jul 18 at 2:59
1
1
Alright. Does that, in this case, mean axioms like the modified version of Euclid's five postulates?
– Tensor McTensorstein
Jul 17 at 10:27
Alright. Does that, in this case, mean axioms like the modified version of Euclid's five postulates?
– Tensor McTensorstein
Jul 17 at 10:27
1
1
I would say so. However I have to say that I am not very familiar with hyperbolic space, much less with the approach used in a particular book.
– Rad80
Jul 17 at 10:41
I would say so. However I have to say that I am not very familiar with hyperbolic space, much less with the approach used in a particular book.
– Rad80
Jul 17 at 10:41
Indeed the hyperbolic plane is a model of Euclid's axioms with the parallel postulate replaced by the postulate that for every line $L$ and every point $p$ not on $L$ there is more than one line through $p$ that is disjoint from $L$.
– Lee Mosher
Jul 18 at 2:59
Indeed the hyperbolic plane is a model of Euclid's axioms with the parallel postulate replaced by the postulate that for every line $L$ and every point $p$ not on $L$ there is more than one line through $p$ that is disjoint from $L$.
– Lee Mosher
Jul 18 at 2:59
add a comment |Â
up vote
1
down vote
There is another notion of "model" which characterizes the hyperbolic plane and is expressed in the language of differential geometry, namely:
The hyperbolic plane is the unique simply connected, complete Riemannian manifold of dimension 2 and constant curvature $-1$.
The meaning of "uniqueness" in this statement is uniqueness up to isometry. More formally, if $mathbb H_1$ and $mathbb H_2$ are two simply connected, complete Riemannian manifolds of dimension 2 and constant curvature $-1$ then there exists an isometry $f : mathbb H_1 to mathbb H_2$.
This notion of uniqueness can be used to prove that all of the models mentioned in your question are isometric to each other, by verifying that the Klein model, the hyperboloid model, and the Poincare model are all complete, simply connected, Riemannian 2-manifolds of constant curvature $-1$.
answered Jul 18 at 3:04
Lee Mosher
45.7k33478
add a comment |Â
up vote
1
down vote
There is another notion of "model" which characterizes the hyperbolic plane and is expressed in the language of differential geometry, namely:
The hyperbolic plane is the unique simply connected, complete Riemannian manifold of dimension 2 and constant curvature $-1$.
The meaning of "uniqueness" in this statement is uniqueness up to isometry. More formally, if $mathbb H_1$ and $mathbb H_2$ are two simply connected, complete Riemannian manifolds of dimension 2 and constant curvature $-1$ then there exists an isometry $f : mathbb H_1 to mathbb H_2$.
This notion of uniqueness can be used to prove that all of the models mentioned in your question are isometric to each other, by verifying that the Klein model, the hyperboloid model, and the Poincare model are all complete, simply connected, Riemannian 2-manifolds of constant curvature $-1$.
answered Jul 18 at 3:04
Lee Mosher
45.7k33478
add a comment |Â
up vote
1
down vote
up vote
1
down vote
There is another notion of "model" which characterizes the hyperbolic plane and is expressed in the language of differential geometry, namely:
The hyperbolic plane is the unique simply connected, complete Riemannian manifold of dimension 2 and constant curvature $-1$.
The meaning of "uniqueness" in this statement is uniqueness up to isometry. More formally, if $mathbb H_1$ and $mathbb H_2$ are two simply connected, complete Riemannian manifolds of dimension 2 and constant curvature $-1$ then there exists an isometry $f : mathbb H_1 to mathbb H_2$.
This notion of uniqueness can be used to prove that all of the models mentioned in your question are isometric to each other, by verifying that the Klein model, the hyperboloid model, and the Poincare model are all complete, simply connected, Riemannian 2-manifolds of constant curvature $-1$.
answered Jul 18 at 3:04
Lee Mosher
45.7k33478
There is another notion of "model" which characterizes the hyperbolic plane and is expressed in the language of differential geometry, namely:
The hyperbolic plane is the unique simply connected, complete Riemannian manifold of dimension 2 and constant curvature $-1$.
The meaning of "uniqueness" in this statement is uniqueness up to isometry. More formally, if $mathbb H_1$ and $mathbb H_2$ are two simply connected, complete Riemannian manifolds of dimension 2 and constant curvature $-1$ then there exists an isometry $f : mathbb H_1 to mathbb H_2$.
This notion of uniqueness can be used to prove that all of the models mentioned in your question are isometric to each other, by verifying that the Klein model, the hyperboloid model, and the Poincare model are all complete, simply connected, Riemannian 2-manifolds of constant curvature $-1$.
answered Jul 18 at 3:04
Lee Mosher
45.7k33478
answered Jul 18 at 3:04
Lee Mosher
45.7k33478
answered Jul 18 at 3:04
Lee Mosher
45.7k33478
answered Jul 18 at 3:04
Lee Mosher
45.7k33478
45.7k33478
add a comment |Â
add a comment |Â
up vote
0
down vote
I would say the word "model" here is used for historical purposes. The history is that people have been wondering whether Euclid's fifth postulate can be proven from the other postulates. This has been shown to be false by constructing a "model" (in the sense of the model theory), i.e., a structure which satisfies the other postulates, but not the fifth postulate. And thus we have models of non-Euclidean geometry.
Comparing with the spherical geometry could be useful. We can view the surface of Earth as a 3D object, or we can make a flat map of Earth using many well-known projections, e.g., stereographic, gnomonic, Mercator. If we apply the stereographic projection to the Minkowski hyperboloid, we get the Poincaré model of the hyperbolic plane. Likewise, gnomonic projection = Klein model, Mercator projection = band model, orthogonal = Gans model. Thus, the word "projection" would be appropriate for some of the models (we have to be very careful though, as the Minkowski hyperboloid lives in the Minkowski space, not the usual $mathbbR^3$) but "model" is still more common. I have written more about this here.
answered Jul 20 at 10:36
Zeno Rogue
33116
add a comment |Â
up vote
0
down vote
I would say the word "model" here is used for historical purposes. The history is that people have been wondering whether Euclid's fifth postulate can be proven from the other postulates. This has been shown to be false by constructing a "model" (in the sense of the model theory), i.e., a structure which satisfies the other postulates, but not the fifth postulate. And thus we have models of non-Euclidean geometry.
Comparing with the spherical geometry could be useful. We can view the surface of Earth as a 3D object, or we can make a flat map of Earth using many well-known projections, e.g., stereographic, gnomonic, Mercator. If we apply the stereographic projection to the Minkowski hyperboloid, we get the Poincaré model of the hyperbolic plane. Likewise, gnomonic projection = Klein model, Mercator projection = band model, orthogonal = Gans model. Thus, the word "projection" would be appropriate for some of the models (we have to be very careful though, as the Minkowski hyperboloid lives in the Minkowski space, not the usual $mathbbR^3$) but "model" is still more common. I have written more about this here.
answered Jul 20 at 10:36
Zeno Rogue
33116
add a comment |Â
up vote
0
down vote
up vote
0
down vote
I would say the word "model" here is used for historical purposes. The history is that people have been wondering whether Euclid's fifth postulate can be proven from the other postulates. This has been shown to be false by constructing a "model" (in the sense of the model theory), i.e., a structure which satisfies the other postulates, but not the fifth postulate. And thus we have models of non-Euclidean geometry.
Comparing with the spherical geometry could be useful. We can view the surface of Earth as a 3D object, or we can make a flat map of Earth using many well-known projections, e.g., stereographic, gnomonic, Mercator. If we apply the stereographic projection to the Minkowski hyperboloid, we get the Poincaré model of the hyperbolic plane. Likewise, gnomonic projection = Klein model, Mercator projection = band model, orthogonal = Gans model. Thus, the word "projection" would be appropriate for some of the models (we have to be very careful though, as the Minkowski hyperboloid lives in the Minkowski space, not the usual $mathbbR^3$) but "model" is still more common. I have written more about this here.
answered Jul 20 at 10:36
Zeno Rogue
33116
I would say the word "model" here is used for historical purposes. The history is that people have been wondering whether Euclid's fifth postulate can be proven from the other postulates. This has been shown to be false by constructing a "model" (in the sense of the model theory), i.e., a structure which satisfies the other postulates, but not the fifth postulate. And thus we have models of non-Euclidean geometry.
Comparing with the spherical geometry could be useful. We can view the surface of Earth as a 3D object, or we can make a flat map of Earth using many well-known projections, e.g., stereographic, gnomonic, Mercator. If we apply the stereographic projection to the Minkowski hyperboloid, we get the Poincaré model of the hyperbolic plane. Likewise, gnomonic projection = Klein model, Mercator projection = band model, orthogonal = Gans model. Thus, the word "projection" would be appropriate for some of the models (we have to be very careful though, as the Minkowski hyperboloid lives in the Minkowski space, not the usual $mathbbR^3$) but "model" is still more common. I have written more about this here.
answered Jul 20 at 10:36
Zeno Rogue
33116
answered Jul 20 at 10:36
Zeno Rogue
33116
answered Jul 20 at 10:36
Zeno Rogue
33116
answered Jul 20 at 10:36
Zeno Rogue
33116
33116
add a comment |Â
add a comment |Â
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