# tilt curve

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

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

are the curves on the surface which intersect (http://planetmath.org/ConvexAngle) 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 fundamental quantities $E,\,F,\,G$ of Gauss, respectively.

 Title tilt curve Canonical name TiltCurve Date of creation 2013-03-22 18:08:22 Last modified on 2013-03-22 18:08:22 Owner pahio (2872) Last modified by pahio (2872) Numerical id 12 Author pahio (2872) Entry type Definition Classification msc 53A05 Classification msc 53A04 Classification msc 51M04 Related topic OrthogonalCurves Related topic Gradient Related topic PositionVector Related topic FirstFundamentalForm Related topic LineOfCurvature