Principal, Asymptotic, and Geodesic Curves

[i1100]

Homework Statement

Is there a curve on a regular surface M that is asymptotic but not principal or geodesic?

Homework Equations

The given definitions of asymptotic, principal, and geodesic:
A principal curve is a curve that is always in a principal direction.
An asymptotic curve is a curve $$\alpha$$ where $$\alpha ''$$ is tangent to M.
A geodesic curve is a curve $$\alpha$$ where $$\alpha ''$$ is normal to M.

The Attempt at a Solution

The closest I have come is finding a curve that is both asymptotic and geodesic, where the example comes on the saddle surface $$z=xy$$ and the curves would be the x and y axes. I just can't think of a surface where there is only an asymptotic curve. This is not a homework assignment, more of a question I have been asking to try to understand the material better.

 

[mighty2000]

I can confirm that it is indeed possible for there to be a curve on a regular surface M that is asymptotic but not principal or geodesic. One example of such a curve is the logarithmic spiral on a sphere. This curve is asymptotic, as its second derivative is always tangent to the surface, but it is not principal or geodesic. Additionally, there are other examples of curves on surfaces that are only asymptotic and not principal or geodesic, such as the curves of constant curvature on a hyperboloid. These curves have a second derivative that is always tangent to the surface, but they are not principal or geodesic.

It is important to note that the definitions of asymptotic, principal, and geodesic curves are based on the curvature of the surface at a given point. Therefore, it is possible for a curve to be asymptotic at some points, but not at others, depending on the curvature of the surface at those points. This is why we can find curves that are asymptotic but not principal or geodesic on certain surfaces.

I hope this helps to clarify your understanding of these concepts. Keep exploring and asking questions, as that is the key to understanding complex concepts in science.