Every finite type curve in R3 is a parabolic-point-free asymptotic line of a suitably constructed plane field, with an explicit hyperbolic closed example given by (sin x, cos x, sin 3x).
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Finite type $\xi$-asymptotic lines of plane fields in $\mathbb{R}^3$
Every finite type curve in R3 is a parabolic-point-free asymptotic line of a suitably constructed plane field, with an explicit hyperbolic closed example given by (sin x, cos x, sin 3x).