3 Examples

The trajectory of a car moving in the plane can be thought of as a curve \([0,2\pi]\to S^1\times \mathbb R^2\). Denoting by \((\varphi,a,c)\) the natural coordinates on \(S^1\times \mathbb R^2\), the angle coordinate \(\varphi\) denotes the orientation of the car with respect to the \(a\)-axis and the coordinates \((a,c)\) the position of the car in the plane. Admissible motions of the car are curves satisfying \[\dot a\sin \varphi = \dot c \cos \varphi.\] The manifold \(S^1\times \mathbb R^2\) together with the contact structure defined by the kernel of the \(1\)-form \(\theta:=\sin \varphi\,\mathrm{d}a - \cos \varphi\, \mathrm{d}c\) is a contact 3-manifold. Indeed, we have \[\theta\wedge\mathrm{d}\theta=-\cos^2\!\varphi\,\mathrm{d}\varphi\wedge\mathrm{d}a \wedge \mathrm{d}c-\sin^2\!\varphi\,\mathrm{d}\varphi\wedge\mathrm{d}a \wedge \mathrm{d}c=-\mathrm{d}\varphi\wedge\mathrm{d}a \wedge \mathrm{d}c\neq 0.\] Applying Theorem 1.1 with \(b=\tan\varphi\) gives an explicit approximation of the curve \[t\mapsto (x(t),y(t),z(t))=(0,0,t).\] Lemma 2.3 gives the loop \[\gamma(t,s)=2(r \cos s,\cos^2s),\] and hence the desired Legendrian curve \[\left(\mathop{\mathrm{arccot}}(r \sec (n t)),2 r t \mathop{\mathrm{sinc}}(n t),t + t\mathop{\mathrm{sinc}}(2 n t)\right),\] provided \(r\) is large enough and \(n\sim r^2\).

The Legendrian approximation of the helix \[\upsilon : [0,2\pi] \to \mathbb R^3, \quad t\mapsto (t,\cos (5t),\sin(5t)),\] with \(n=\frac{2}{9}r^2\) and \(r=30\) is given by \[\begin{aligned} a(t) =& ~ t + \frac{3}{20} \sin(200 t)\\ b(t) =&~ \frac{455}{451} \cos(5 t) +\frac{120}{451}\cos(5t) \cos(200 t) \\ c(t) =&~ \sin(5 t) + \frac{459}{5863} \sin(195 t) + \frac{1377}{18491} \sin(205 t)+\frac{180}{35629} \sin(395 t)+\\ &~ + \frac{20}{4059}\sin(405 t).\end{aligned}\] and produces the zig-zags and the small loops in its front and Lagrangian projections (see Figure 1).