tilt curve

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


are the curves on the surface which intersect (http://planetmath.org/ConvexAngle) orthogonally the level curvesMathworldPlanetmathf(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=±r2-x2-y2  are the “latitude circles”, the tilt curves of the sphere are the “meridianMathworldPlanetmath circles”.  The tilt curves of a helicoid are circular helices.

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

dydx=fy(x,y)fx(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


The level curves are the ellipsesPlanetmathPlanetmathx2a2+y2b2=c.  Now we have


whence the differential equation of the tilt curves is


The separation of variablesMathworldPlanetmath and the integration yield




and finally

y=C|x|a2/b2. (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


of 3, e.g. in the form


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

dvdu=-g11+g12f(u,v)g12+g22f(u,v), (3)



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