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 curves of the surface. If the gravitation acts in direction of the negative -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 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 -plane, the differential equation of those projection curves is
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 ellipses . Now we have
whence the differential equation of the tilt curves is
The separation of variables and the integration yield
Here, we may allow for 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 , e.g. in the form
the family of its orthogonal curves on the surface has in the Gaussian coordinates the differential equation
are the fundamental quantities of Gauss, respectively.
|Date of creation||2013-03-22 18:08:22|
|Last modified on||2013-03-22 18:08:22|
|Last modified by||pahio (2872)|