Parallels and meridians on a pseudosphere (tractricoid)
I am trying to visualize and understand parallels and meridians on a pseudosphere with respect to the usual parallels and meridians on the sphere.
On a sphere of radius $r$, using the usual $theta, phi $ (polar angle, azimuthal angle) parameterisation, we find that the parallels are the $theta = rm{const.}$ curves, and the meridians are the $phi = rm{const.}$ curves. Moreover, the length of the meridians is $2pi r$ whereas the length of a parallel is $r cos{theta}$.
Are there equivalent formulae for the 'parallels' and 'meridians' on the pseudosphere (I mean the lines drawn here: http://mathworld.wolfram.com/images/eps-gif/Pseudosphere_700.gif)? The meridians are infinitely long, but can the parallels be written as the $cos$ or $cosh$ of some equivalent of the $theta$ parameter, in some parameterization?
geometry hyperbolic-geometry noneuclidean-geometry
asked Nov 30 '18 at 2:37
ap21ap21
385
add a comment |
I am trying to visualize and understand parallels and meridians on a pseudosphere with respect to the usual parallels and meridians on the sphere.
On a sphere of radius $r$, using the usual $theta, phi $ (polar angle, azimuthal angle) parameterisation, we find that the parallels are the $theta = rm{const.}$ curves, and the meridians are the $phi = rm{const.}$ curves. Moreover, the length of the meridians is $2pi r$ whereas the length of a parallel is $r cos{theta}$.
Are there equivalent formulae for the 'parallels' and 'meridians' on the pseudosphere (I mean the lines drawn here: http://mathworld.wolfram.com/images/eps-gif/Pseudosphere_700.gif)? The meridians are infinitely long, but can the parallels be written as the $cos$ or $cosh$ of some equivalent of the $theta$ parameter, in some parameterization?
geometry hyperbolic-geometry noneuclidean-geometry
asked Nov 30 '18 at 2:37
ap21ap21
385
1
The term “pseudosphere” is somewhat overloaded: different people associate different meaning with it. Judging from your picture, I'd say you are speaking about a tractricoid. Might be useful to says so in the text.
– MvG
Dec 1 '18 at 23:49
add a comment |
I am trying to visualize and understand parallels and meridians on a pseudosphere with respect to the usual parallels and meridians on the sphere.
On a sphere of radius $r$, using the usual $theta, phi $ (polar angle, azimuthal angle) parameterisation, we find that the parallels are the $theta = rm{const.}$ curves, and the meridians are the $phi = rm{const.}$ curves. Moreover, the length of the meridians is $2pi r$ whereas the length of a parallel is $r cos{theta}$.
Are there equivalent formulae for the 'parallels' and 'meridians' on the pseudosphere (I mean the lines drawn here: http://mathworld.wolfram.com/images/eps-gif/Pseudosphere_700.gif)? The meridians are infinitely long, but can the parallels be written as the $cos$ or $cosh$ of some equivalent of the $theta$ parameter, in some parameterization?
geometry hyperbolic-geometry noneuclidean-geometry
asked Nov 30 '18 at 2:37
ap21ap21
385
I am trying to visualize and understand parallels and meridians on a pseudosphere with respect to the usual parallels and meridians on the sphere.
On a sphere of radius $r$, using the usual $theta, phi $ (polar angle, azimuthal angle) parameterisation, we find that the parallels are the $theta = rm{const.}$ curves, and the meridians are the $phi = rm{const.}$ curves. Moreover, the length of the meridians is $2pi r$ whereas the length of a parallel is $r cos{theta}$.
Are there equivalent formulae for the 'parallels' and 'meridians' on the pseudosphere (I mean the lines drawn here: http://mathworld.wolfram.com/images/eps-gif/Pseudosphere_700.gif)? The meridians are infinitely long, but can the parallels be written as the $cos$ or $cosh$ of some equivalent of the $theta$ parameter, in some parameterization?
geometry hyperbolic-geometry noneuclidean-geometry
geometry hyperbolic-geometry noneuclidean-geometry
asked Nov 30 '18 at 2:37
ap21ap21
385
asked Nov 30 '18 at 2:37
ap21ap21
385
edited Dec 2 '18 at 3:27
ap21
asked Nov 30 '18 at 2:37
ap21ap21
385
asked Nov 30 '18 at 2:37
ap21ap21
385
asked Nov 30 '18 at 2:37
ap21ap21
385
385
1
The term “pseudosphere” is somewhat overloaded: different people associate different meaning with it. Judging from your picture, I'd say you are speaking about a tractricoid. Might be useful to says so in the text.
– MvG
Dec 1 '18 at 23:49
add a comment |
1
The term “pseudosphere” is somewhat overloaded: different people associate different meaning with it. Judging from your picture, I'd say you are speaking about a tractricoid. Might be useful to says so in the text.
– MvG
Dec 1 '18 at 23:49
1
1
The term “pseudosphere” is somewhat overloaded: different people associate different meaning with it. Judging from your picture, I'd say you are speaking about a tractricoid. Might be useful to says so in the text.
– MvG
Dec 1 '18 at 23:49
The term “pseudosphere” is somewhat overloaded: different people associate different meaning with it. Judging from your picture, I'd say you are speaking about a tractricoid. Might be useful to says so in the text.
– MvG
Dec 1 '18 at 23:49
add a comment |
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The term “pseudosphere” is somewhat overloaded: different people associate different meaning with it. Judging from your picture, I'd say you are speaking about a tractricoid. Might be useful to says so in the text.
– MvG
Dec 1 '18 at 23:49