Remark 4.6 . [0463]

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Remark 4.6.

The trigonometric factors in $K_{p}(a)$ have elementary geometric interpretations. The Euclidean metric $g_{a}^{\prime}$ induces an inner product on $\mathbb{R}^{2}_{y_{1},y_{2}}$ . Then the angles between the asymptotic directions of $S$ are

\begin{cases}\angle(\mathfrak{D}_{1},\mathfrak{D}_{3})=\frac{\pi}{2}+\arctan(\frac{a_{2\bar{2}}+\text{Re}(a_{1\bar{2}})}{\sqrt{\mathbb{A}}}),\\ \angle(\mathfrak{D}_{2},\mathfrak{D}_{3})=\frac{\pi}{2}+\arctan(\frac{a_{1\bar{1}}+\text{Re}(a_{1\bar{2}})}{\sqrt{\mathbb{A}}}).\end{cases}