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

${\displaystyle \frac{dy}{dx}}={\displaystyle \frac{f_y^{\prime }(x,y)}{f_x^{\prime }(x,y)}}.$(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

$$f_x^{\prime }(x,y)=\frac{\partial }{\partial x}\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)=\frac{2x}{a^2},f_y^{\prime }(x,y)=\frac{\partial }{\partial y}\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)=\frac{2y}{b^2},$$

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

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

$$\overrightarrow{r}={(x(u,v),y(u,v),z(u,v))}^{⊺}$$

of ${ℝ}^3$, 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

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

where

$$g_{11}={\overrightarrow{r}}_u^{\prime }\cdot {\overrightarrow{r}}_u^{\prime },g_{12}={\overrightarrow{r}}_u^{\prime }\cdot {\overrightarrow{r}}_v^{\prime },g_{22}={\overrightarrow{r}}_v^{\prime }\cdot {\overrightarrow{r}}_v^{\prime }$$

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