Me and my classmates are interested in a visual description of an asymptotic direction at a point of a surface. The normal curvature in an asymptotic direction at a point is zero. And a curve on a surface is called asymptotic if its tangent is parallell to an asymptotic direction at each point.
The covariant derivative of the tangent vector field of an asymptotic curve along the same should equal the second derivative of the same curve, sine the normal component is zero. So the osculating plane and the tangent plane coincide for asymptotic curves.
I.e the first and second order derivative of the curve both belongs to the tangent space. This makes me think of the curve as "being intrinsic", never bending away from the surface, but how can we visualize this? The only visual understanding we have is the example of a curve in a plane.
Why is it called asymptotic?
Maybe someone good at mathematical drawing cold provide some nice pictures.
Thanks for any comments, examples or drawings!
Best Answer
I belong to the old school believing that experience comes first, explanation comes next. To believe fully the object should be held in the hands.
For a normal curvature to vanish its neighboring planes should have normal curvatures of alternating sign. Accordingly real asymptotic directions can exist only on negative Gauss curvature K surfaces. That is, only around saddle points.
For an intuitive grasp please pick up any such surface ( K <0) you can get. Around any arbitrary saddle point rotate the straight (steel) edge of a ruler all the 360 degrees in a normal plane.
You will notice that two vertically opposite hills do not permit rotation and you are free to rotate on the rest of the sloping down area. The demarcating position of ruler edge lines are the traces of local asymptotic lines.
Vertical sections around a saddle point are a series of hyperbolas. (Physically one can see them as interference fringes for monochromatic light in an Optics lab or as random but hyperbolically ordered scratches on an automobile front engine cover.) The lines of zero normal curvature are common asymptotes of this hyperbola set. They can also be understood through standard Dupin's indicatrices, the level curves of intersection.
Imho Mohr circle is the best way to depict the $ \kappa_n = 0 $ positioning in a drawing.
To visualize them look at pictures of loaded 3D fishnet as asymptotic lines of negative K .. which is an isometry invariant. Also geodesic torsion of these lines intrinsically equals square root of (-K). ( Enneper-Beltrami thm).
EDIT1:
Euler's relation of normal curvatures as you rotate the steel edge around a saddle point until obstructed by an asymptotic direction limit of the tangent bundle :
$$ \kappa_1 \cos^2 \alpha + \kappa_2 \sin^2 \alpha = \kappa_n = 0 $$
$$ \tau_g^2 = - k_1 \cdot k_2 $$
shown on the Mohr's circle:
EDIT2:
What you may have intuitively expected in isometry conservation is normal curvature in tangent plane but not geodesic curvature. It is the other way round.
During bending by virtue of Egregium theorem geodesic curvature does not change but normal curvature that was once zero for an asymptotic line undergoes a change, i.e, loses its asymptotic character.
So it is recommended to watch the animation several times:
https://en.wikipedia.org/wiki/Helicoid
During bending the asymptotes migrate quite a lot.. by rotating around each saddle point. In fact asymptotes become geodesics and vice-versa. In engineering, it is described as non-linear deformation.