  In this paper, we regard the smooth quadric threefold $Q_{3}$ as Lagrangian
Grassmannian and search for fixed rational curves of low degree in $Q_{3}$ with
respect to a torus action, which is the maximal subgroup of the special linear
group $\text{SL}(2,\mathbb{C})$. Most of them are confirmations of very
well-known facts. If the degree of a rational curve is $3$, it is confirmed
using the Lagrangian's geometric properties that the moduli space of twisted
cubic curves in $Q_3$ has a specific projective bundle structure. From this, we
can immediately obtain the cohomology ring of the moduli space.
