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The topic is quite old.

There are three and only three types of rotationally symmetric surfaces for constant K = -1/a^2 where a is the cuspidal radius of the central pseudosphere. These are the central pseudosphere (parameter m =1) or tractricoid  , Conic type ( m < 1) and Ring type ( m > 1). The descriptions Ring, Conic type etc. are given by Klein in:

Felix Klein," Vorlesungen über nicht-euklidische Geometrie" 3rd ed. (Berlin, 1928). with a reference/reprint iirc from Crelle's Journal.

Asymptotic lines of a Chebychev Net on these three surfaces are given by the Sine-Gordon Equation.

Present day English translations of Vorlesungen may be available on googling, else one can contact German newsgroups e.g., de.sci.mathematik.

Although the central pseudosphere is often referred to as "the" Beltrami surface, the physical paper model he made is isometrically equivalent to the Conic type ( m > 1) that you can readily verify in Daina's blog. I am discussing it the concurrent thread here

In 1868 when making his hyperbolic plane model Beltrami was a professor of mathematics in Bologna. It had to be important to him to take his paper models with him when Beltrami returned to University of Pavia 1876

The topic is quite old.

There are three and only three types of rotationally symmetric surfaces for constant $K = -1/a^2$ where $a$ is the cuspidal radius of the central pseudosphere. These are the central pseudosphere (parameter $m =1$) or tractricoid, Conic type ($m < 1$) and Ring type ($m > 1$). The descriptions Ring, Conic type etc. are given by Klein in:

Felix Klein," Vorlesungen über nicht-euklidische Geometrie" 3rd ed. (Berlin, 1928). with a reference/reprint iirc from Crelle's Journal.

Asymptotic lines of a Chebyshev Net on these three surfaces are given by the Sine-Gordon Equation.

Present day English translations of Vorlesungen may be available on googling, else one can contact German newsgroups e.g., de.sci.mathematik.

Although the central pseudosphere is often referred to as "the" Beltrami surface, the physical paper model he made is isometrically equivalent to the Conic type ( m > 1) that you can readily verify in Daina's blog. I am discussing it the concurrent thread here

In 1868 when making his hyperbolic plane model Beltrami was a professor of mathematics in Bologna. It had to be important to him to take his paper models with him when Beltrami returned to University of Pavia 1876.

The topic is quite old.

There are three and only three types of rotationally symmetric surfaces for constant K = -1/a^2 where a is the cuspidal radius of the central pseudosphere. These are the central pseudosphere (parameter m =1) or tractricoid , Conic type ( m < 1) and Ring type ( m > 1). The descriptions Ring, Conic type etc. are given by Klein in:

Felix Klein," Vorlesungen über nicht-euklidische Geometrie" 3rd ed. (Berlin, 1928). with a reference/reprint iirc from Crelle's Journal.

Asymptotic lines of a Chebychev Net on these three surfaces are given by the Sine-Gordon Equation.

Present day English translations of Vorlesungen may be available on googling, else one can contact German newsgroups e.g., de.sci.mathematik.

Although the central pseudosphere is often referred to as "the" Beltrami surface, the physical paper model he made is isometrically equivalent to the Conic type ( m > 1) that you can readily verify in Daina's blog. I am discussing it the concurrent thread here

In 1868 when making his hyperbolic plane model Beltrami was a professor of mathematics in Bologna. It had to be important to him to take his paper models with him when Beltrami returned to University of Pavia 1876

The topic is quite old.

There are three and only three types of rotationally symmetric surfaces for constant K = -1/a^2 where a is the cuspidal radius of the central pseudosphere. These are the central pseudosphere (parameter m =1) or tractricoid , Conic type ( m < 1) and Ring type ( m > 1). The descriptions Ring, Conic type etc. are given by Klein in:

Felix Klein," Vorlesungen über nicht-euklidische Geometrie" 3rd ed. (Berlin, 1928). with a reference/reprint iirc from Crelle's Journal.

Asymptotic lines of a Chebychev Net on these three surfaces are given by the Sine-Gordon Equation.

Present day English translations of Vorlesungen may be available on googling, else one can contact German newsgroups e.g., de.sci.mathematik.

Although the central pseudosphere is often referred to as "the" Beltrami surface, the physical paper model he made is isometrically equivalent to the Conic type ( m > 1) that you can readily verify in Daina's blog. I am discussing it the concurrent thread here

In 1868 when making his hyperbolic plane model Beltrami was a professor of mathematics in Bologna. It had to be important to him to take his paper models with him when Beltrami returned to University of Pavia 1876

The topic is quite old.

There are three only three types of rotationally symmetric surfaces for constant K = -1/a^2 where a is the cuspidal radius of the central pseudosphere. These are the central pseudosphere (parameter m =1) or tractricoid , Conic type ( m < 1) and Ring type ( m > 1). The descriptions Ring, Conic type etc. are given by Klein in:

Felix Klein," Vorlesungen über nicht-euklidische Geometrie" 3rd ed. (Berlin, 1928). with a reference/reprint iirc from Crelle's Journal.

Asymptotic lines of a Chebychev Net are given by the Sine-Gordon Equation.

Present day English translations of Vorlesungen may be available on googling, else one can contact German newsgroups e.g., de.sci.mathematik.

Although the central pseudosphere is often referred to as "the" Beltrami surface, the physical paper model he made is isometrically equivalent to the Conic type ( m > 1) that you can readily verify in Daina's blog. I am discussing it the concurrent thread here

In 1868 when making his hyperbolic plane model Beltrami was a professor of mathematics in Bologna. It had to be important to him to take his paper models with him when Beltrami returned to University of Pavia 1876

The topic is quite old.

There are three and only three types of rotationally symmetric surfaces for constant K = -1/a^2 where a is the cuspidal radius of the central pseudosphere. These are the central pseudosphere (parameter m =1) or tractricoid , Conic type ( m < 1) and Ring type ( m > 1). The descriptions Ring, Conic type etc. are given by Klein in:

Felix Klein," Vorlesungen über nicht-euklidische Geometrie" 3rd ed. (Berlin, 1928). with a reference/reprint iirc from Crelle's Journal.

Asymptotic lines of a Chebychev Net on these three surfaces are given by the Sine-Gordon Equation.

Present day English translations of Vorlesungen may be available on googling, else one can contact German newsgroups e.g., de.sci.mathematik.

Although the central pseudosphere is often referred to as "the" Beltrami surface, the physical paper model he made is isometrically equivalent to the Conic type ( m > 1) that you can readily verify in Daina's blog. I am discussing it the concurrent thread here

In 1868 when making his hyperbolic plane model Beltrami was a professor of mathematics in Bologna. It had to be important to him to take his paper models with him when Beltrami returned to University of Pavia 1876