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# orthogonal curve

If a family of plane curves (with one free parameter) satisfies the differential equation

$F(x,\,y,\,y^{{\prime}})\;=\;0,$ |

where $y^{{\prime}}=\frac{dy}{dx}$, then the family of curves intersecting orthogonally all the first curves satisfies the differential equation

$F\left(x,\,y,\,-\frac{1}{y^{{\prime}}}\right)\;=\;0.$ |

Anyone of the latter curves is an *orthogonal curve* of the former ones.

Example. Let’s consider the family of rectangular hyperbolas

$x^{2}-y^{2}\;=\;c$ |

with the parameter $c$ taking any real value. Derivating with respect to $x$ gives the differential equation of this family,

$x-yy^{{\prime}}\;=\;0,$ |

and by replacing here $y^{{\prime}}$ with $-\frac{1}{y^{{\prime}}}$ we obtain the differential equation

$x+\frac{y}{y^{{\prime}}}\;=\;0$ |

of the orthogonal curves. Integrating its form

$\frac{dy}{y}\;=\;-\frac{dx}{x}$ |

gives the solution

$xy\;=\;C,$ |

which represents another family of rectangular hyperbolas.

In the picture below (by drini), there are four hyperbolas of the first family (blue) given by the values $c=-1,\,-2,\,-4,\,-8$ and four hyperbolas of the orthogonal family (red) given by the values $C=1,\,2,\,4,\,8$.

## Mathematics Subject Classification

34C99*no label found*34C05

*no label found*

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