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Hometilt curve

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# tilt curve

The tilt curves (in German die Neigungskurven) of a surface

$z=f(x,\,y)$ |

are the curves on the surface which intersect orthogonally the level curves $f(x,\,y)=c$ of the surface. If the gravitation acts in direction of the negative $z$-axis, then a drop of water on the surface aspires to slide along a tilt curve. For example, since the level curves of the sphere $z=\pm\sqrt{r^{2}-x^{2}-y^{2}}$ are the “latitude circles”, the tilt curves of the sphere are the “meridian circles”. The tilt curves of a helicoid are circular helices.

If the tilt curves are projected on the $xy$-plane, the differential equation of those projection curves is

$\displaystyle\frac{dy}{dx}=\frac{f^{{\prime}}_{y}(x,\,y)}{f^{{\prime}}_{x}(x,% \,y)}.$ | (1) |

Naturally, they also cut orthogonally (the projections of) the level curves.

Example. Let us find the tilt curves of the elliptic paraboloid

$z=\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}.$ |

The level curves are the ellipses $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=c$. Now we have

$f^{{\prime}}_{x}(x,\,y)=\frac{\partial}{\partial x}\!\left(\frac{x^{2}}{a^{2}}% \!+\!\frac{y^{2}}{b^{2}}\right)=\frac{2x}{a^{2}},\quad f^{{\prime}}_{y}(x,\,y)% =\frac{\partial}{\partial y}\!\left(\frac{x^{2}}{a^{2}}\!+\!\frac{y^{2}}{b^{2}% }\right)=\frac{2y}{b^{2}},$ |

whence the differential equation of the tilt curves is

$\frac{dy}{dx}=\frac{a^{2}}{b^{2}}\!\cdot\!\frac{y}{x}.$ |

The separation of variables and the integration yield

$\int\frac{dy}{y}=\frac{a^{2}}{b^{2}}\!\int\frac{dx}{x},$ |

then

$\ln|y|=\frac{a^{2}}{b^{2}}\ln|x|+\ln|C|=\ln(|C||x|^{{a^{2}/b^{2}}}),$ |

and finally

$\displaystyle y=C|x|^{{a^{2}/b^{2}}}.$ | (2) |

Here, we may allow for $C$ all positive and negative values. The curves (2) originate from the origin and continue infinitely far.

Remark. Given an arbitrary family of parametre curves on a surface

$\vec{r}\,=\,(x(u,\,v),\;y(u,\,v),\;z(u,\,v))^{\intercal}$ |

of $\mathbb{R}^{3}$, e.g. in the form

$\frac{du}{dv}=f(u,\,v),$ |

the family of its orthogonal curves on the surface has in the Gaussian coordinates $u,\,v$ the differential equation

$\displaystyle\frac{dv}{du}=-\frac{g_{{11}}+g_{{12}}f(u,\,v)}{g_{{12}}+g_{{22}}% f(u,\,v)},$ | (3) |

where

$g_{{11}}=\vec{r}\,^{{\prime}}_{u}\cdot\vec{r}\,^{{\prime}}_{u},\quad g_{{12}}=% \vec{r}\,^{{\prime}}_{u}\cdot\vec{r}\,^{{\prime}}_{v},\quad g_{{22}}=\vec{r}\,% ^{{\prime}}_{v}\cdot\vec{r}\,^{{\prime}}_{v}$ |

are the first order fundamental quantities $E,\,F,\,G$ of Gauss, respectively.

## Mathematics Subject Classification

53A05*no label found*53A04

*no label found*51M04

*no label found*

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## Comments

## "Tilt curve"

I have chosen the name "tilt curve", but it's probable that it is not correct. Is there some old official term for that concept?

## Re: "Tilt curve"

Well pahio, I didn't know such a denomination but I suspect there are several names to designate those "tilt curves". I think depending on the matter. For examples:

- In fluid mechanics you have the potential of velocities $\phi(x,y,z)=C$, similar to level surfaces composed by level curves, and the stream-function $\psi(x,y,z)=K$ composed by the stramlines (velocity field is tangent to them).

- In PDE theory those "tilt functions" are named characteristics, which are essential to study the behavior (in general locally) of the PDE.

But for sure there are a lot of another applications, for instance, in theoretical Physics.

peruchin