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There are many examples of surfaces in $\mathbb{R}^3$ with constant negative curvature. They can be described by using the so-called parametrization by Chebyshev nets. Have a look at the paper by Robert McLachlan

A gallery of constant-negative-curvature surfaces, The Mathematical Intelligencer 16 (1994), 31-37.

However (and this answers your question) there are precisely three types of revolution surfaces with constant curvature $-1$. The reason is explained here (it boils down to solving a second order ODE):

http://math.stackexchange.com/questions/77396/revolution-surfaces-of-constant-gaussian-curvaturehttps://math.stackexchange.com/questions/77396/revolution-surfaces-of-constant-gaussian-curvature

One of them of course is the tracioid. Pictures of the three surfaces can be found here:

http://demonstrations.wolfram.com/SurfacesOfRevolutionWithConstantGaussianCurvature/

There are many examples of surfaces in $\mathbb{R}^3$ with constant negative curvature. They can be described by using the so-called parametrization by Chebyshev nets. Have a look at the paper by Robert McLachlan

A gallery of constant-negative-curvature surfaces, The Mathematical Intelligencer 16 (1994), 31-37.

However (and this answers your question) there are precisely three types of revolution surfaces with constant curvature $-1$. The reason is explained here (it boils down to solving a second order ODE):

http://math.stackexchange.com/questions/77396/revolution-surfaces-of-constant-gaussian-curvature

One of them of course is the tracioid. Pictures of the three surfaces can be found here:

http://demonstrations.wolfram.com/SurfacesOfRevolutionWithConstantGaussianCurvature/

There are many examples of surfaces in $\mathbb{R}^3$ with constant negative curvature. They can be described by using the so-called parametrization by Chebyshev nets. Have a look at the paper by Robert McLachlan

A gallery of constant-negative-curvature surfaces, The Mathematical Intelligencer 16 (1994), 31-37.

However (and this answers your question) there are precisely three types of revolution surfaces with constant curvature $-1$. The reason is explained here (it boils down to solving a second order ODE):

https://math.stackexchange.com/questions/77396/revolution-surfaces-of-constant-gaussian-curvature

One of them of course is the tracioid. Pictures of the three surfaces can be found here:

http://demonstrations.wolfram.com/SurfacesOfRevolutionWithConstantGaussianCurvature/

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Francesco Polizzi
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There are many examples of surfaces in $\mathbb{R}^3$ with constant negative curvature. They can be described by using the so-called parametrization by Chebyshev nets. Have a look at the paper by Robert McLachlan

A gallery of constant-negative-curvature surfaces, The Mathematical Intelligencer 16 (1994), 31-37.

However (and this answeranswers your question) there are precisely three types of revolution surfaces with constant curvature $-1$. The reason is explained here (it boils down to solving a second order ODE):

http://math.stackexchange.com/questions/77396/revolution-surfaces-of-constant-gaussian-curvature

One of them of course is the tracioid. Pictures of the three surfaces can be found here:

http://demonstrations.wolfram.com/SurfacesOfRevolutionWithConstantGaussianCurvature/

There are many examples of surfaces in $\mathbb{R}^3$ with constant negative curvature. They can be described by using the so-called parametrization by Chebyshev nets. Have a look at the paper by Robert McLachlan

A gallery of constant-negative-curvature surfaces, The Mathematical Intelligencer 16 (1994), 31-37.

However (and this answer your question) there are precisely three types of revolution surfaces with constant curvature $-1$. The reason is explained here (it boils down to solving a second order ODE):

http://math.stackexchange.com/questions/77396/revolution-surfaces-of-constant-gaussian-curvature

One of them of course is the tracioid. Pictures of the three surfaces can be found here:

http://demonstrations.wolfram.com/SurfacesOfRevolutionWithConstantGaussianCurvature/

There are many examples of surfaces in $\mathbb{R}^3$ with constant negative curvature. They can be described by using the so-called parametrization by Chebyshev nets. Have a look at the paper by Robert McLachlan

A gallery of constant-negative-curvature surfaces, The Mathematical Intelligencer 16 (1994), 31-37.

However (and this answers your question) there are precisely three types of revolution surfaces with constant curvature $-1$. The reason is explained here (it boils down to solving a second order ODE):

http://math.stackexchange.com/questions/77396/revolution-surfaces-of-constant-gaussian-curvature

One of them of course is the tracioid. Pictures of the three surfaces can be found here:

http://demonstrations.wolfram.com/SurfacesOfRevolutionWithConstantGaussianCurvature/

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Francesco Polizzi
  • 66.3k
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There are actually many examples of themsurfaces in $\mathbb{R}^3$ with constant negative curvature. They can be described by using the so-called parametrization by Chebyshev nets. Have a look at the paper by Robert McLachlan

A gallery of constant-negative-curvature surfaces, The Mathematical Intelligencer 16 (1994), 31-37.

However (and this answer your question) there are precisely three types of revolution surfaces with constant curvature $-1$. The reason is explained here (it boils down to solving a second order ODE):

http://math.stackexchange.com/questions/77396/revolution-surfaces-of-constant-gaussian-curvature

One of them of course is the tracioid. Pictures of the three surfaces can be found here:

http://demonstrations.wolfram.com/SurfacesOfRevolutionWithConstantGaussianCurvature/

There are actually many of them. They can be described by using the so-called parametrization by Chebyshev nets. Have a look at the paper by Robert McLachlan

A gallery of constant-negative-curvature surfaces, The Mathematical Intelligencer 16 (1994), 31-37.

There are many examples of surfaces in $\mathbb{R}^3$ with constant negative curvature. They can be described by using the so-called parametrization by Chebyshev nets. Have a look at the paper by Robert McLachlan

A gallery of constant-negative-curvature surfaces, The Mathematical Intelligencer 16 (1994), 31-37.

However (and this answer your question) there are precisely three types of revolution surfaces with constant curvature $-1$. The reason is explained here (it boils down to solving a second order ODE):

http://math.stackexchange.com/questions/77396/revolution-surfaces-of-constant-gaussian-curvature

One of them of course is the tracioid. Pictures of the three surfaces can be found here:

http://demonstrations.wolfram.com/SurfacesOfRevolutionWithConstantGaussianCurvature/

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Francesco Polizzi
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Francesco Polizzi
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