Parametric breather surfaces are known in one-to-one correspondence with the solutions of a certain non-linear wave-equation, i.e., the so-called Sine-Gordon Equation. It turns out, solutions to this equation correspond to unique pseudospherical surfaces, namely soliton. Breather surface corresponds to a time-periodic 2-soliton solution.

Parametric breather surface has the following parametric equations :

Where , controls how far the tip goes, and controls the girth.

When , , and :

With orthographic projection :

**About Pseudospherical Surfaces :**

Surface in having constant Gaussian curvature are usually called *pseudospherical surfaces*.

If is a surface with Gaussian curvature then it is known that there exists a local asymptotic coordinate system on such that the first and second fundamental forms are:

, and ,

where is the angle between asymptotic lines (the *x*-curves and *t*-curves). The Gauss-Codazzi equations for in these coordinates become a single equation, the sine-Gordon equation (SGE) :

The SGE is one of the model soliton equations.

**References and Readings :**

- Chuu-Lian Terng. 2004.
*Lecture notes on curves and surfaces in* , available here.
- Chuu-Lian Terng. 1990s.
*About Pseudospherical Surfaces*, available here.
- Richard S Palais. 2003.
*A Modern Course on Curves and Surfaces*, available here.

**About 3D-XplorMath :**

3D-XplorMath is a Mathematical Visualization program. The older original version, written in Pascal, runs only on Macintosh computers, but there is also a newer cross-platform Java version, called 3D-XplorMath-J which is written in the Java programming language; to use it, you must have Java 5.0 or higher installed on your computer.

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