# How is the frenet frame along an asymptotic curve related to the geometry of the surface?

I'm reading Differential Geometry: A first course in curves and surfaces by Theodore Shifrin and here is one of the questions from the exercise. I just can't seem to make the connection between the two.

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• Mathematics not statistics? – Nick Cox Apr 23 at 8:08
• Welcome to MSE. It's preferred form to put the actual question in the question, not the title. :) Also, you should give some indication about what you've tried. Given that I wrote the text, I have some idea of what tools you have available :) – Ted Shifrin Apr 24 at 16:55

HINT: Assuming $$\kappa\ne 0$$, so that the Frenet frame actually is defined, if the curve is an asymptotic curve, what is $$\kappa\mathbf N\cdot\bf n$$? (Here $$\mathbf N$$ is the principal normal and $$\mathbf n$$ is the surface normal.)

• will $\kappa\mathbf N\cdot\bf n=\kappa$ because both the normal unit vectors are in the same direction? – Mathematical Mushroom Apr 25 at 16:49
• @MathematicalMushroom Why do you say that? What is the definition of an asymptotic curve, by the way? – Ted Shifrin Apr 25 at 16:51
• An asymptotic curve is always tangent to the asymptotic direction, which is the direction in which the normal curvature is zero. So if the normal curvature of a surface in that direction is zero, and a curve is on the surface going in that direction, then the normal unit vector of the curve must also not be changing. – Mathematical Mushroom Apr 25 at 16:53
• @MathematicalMushroom No, that last statement is wrong. Yes, normal curvature is $0$. So what does that have to do with the question I posed? – Ted Shifrin Apr 25 at 16:54
• perpendicular!! – Mathematical Mushroom Apr 25 at 19:29