# Homework Help: Principal, Asymptotic, and Geodesic Curves

1. Apr 9, 2010

### i1100

1. The problem statement, all variables and given/known data
Is there a curve on a regular surface M that is asymptotic but not principal or geodesic?

2. Relevant 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.

3. 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.