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.

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