# equal arc length and area

We want to determine the nonnegative differentiable^{} real functions $x\mapsto y$ whose graph has the property that the arc length^{} between any two points of it is the same as the area (http://planetmath.org/AreaOfPlaneRegion) by the curve, the $x$-axis and the ordinate lines of those points.

The requirement leads to the equation

${\int}_{a}^{x}}\sqrt{1+{\left({\displaystyle \frac{dy}{dx}}\right)}^{2}}\mathit{d}x={\displaystyle {\int}_{a}^{x}}y\mathit{d}x.$ | (1) |

By the fundamental theorem of calculus^{}, we infer from (1) the differential equation^{}

$\sqrt{1+{\left({\displaystyle \frac{dy}{dx}}\right)}^{2}}=y,$ | (2) |

whence $\frac{dy}{dx}=\sqrt{{y}^{2}-1}$. In the case $y\not\equiv 1$, the separation of variables^{} yields

$$\int \mathit{d}x=\int \frac{dy}{\sqrt{{y}^{2}-1}},$$ |

i.e.

$$x+C=\mathrm{arcosh}y.$$ |

Consequently, the equation (2) has the general solution

$y=\mathrm{cosh}(x+C)$ | (3) |

and the singular solution

$y\equiv \mathrm{\hspace{0.33em}1}.$ | (4) |

The functions defined by (3) and (4) are the only satisfying the given requirement. The graphs are a chain curve (which may be translated in the horizontal direction) and a line parallel^{} to the $x$-axis. Evidently, the line is the envelope of the integral curves given be the general solution.

Title | equal arc length and area |

Canonical name | EqualArcLengthAndArea |

Date of creation | 2013-03-22 19:13:36 |

Last modified on | 2013-03-22 19:13:36 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 8 |

Author | pahio (2872) |

Entry type | Example |

Classification | msc 53A04 |

Classification | msc 34A34 |

Classification | msc 34A05 |

Classification | msc 26A09 |

Synonym | equal area and arc length |

Related topic | Arcosh |

Related topic | HyperbolicFunctions |

Related topic | ChainCurve |