Let $S^1=\mathbb R^1/\mathbb Z$. Consider a family $\varphi_t$ of pieceswise smooth injective maps $\varphi_t:S^1\to \mathbb C^1$ depending continuously on $t$. Then each curve $\varphi_t(S^1)$ is a simple closed curve in $\mathbb C^1$, i.e. it bounds an open complex disk. Using Riemann mapping theorem we can identify such a disk with the unit disk $|z|\le 1$. In particular, for each $t$ we can define the cross-ratio of points $\varphi_t(0)$, $\varphi_t(\frac{1}{4})$, $\varphi_t(\frac{1}{2})$, $\varphi_t(\frac{3}{4})$ as points lying on the boundary of the disk bounded by $\varphi(t)$. I am pretty sure that such a cross-ratio is a continuous function of $t$.

Question. Does this statement follow from some standard result? How to prove it?

Let $S^1=\mathbb R^1/\mathbb Z$. Consider a family $\varphi_t$ of pieceswise smooth injective maps $\varphi_t:S^1\to \mathbb C^1$ depending continuously on $t$. Then each curve $\varphi_t(S^1)$ is a simple closed curve in $\mathbb C^1$, i.e. it bounds an open complex disk. Using Riemann mapping theorem we can identify such a disk with the unit disk $|z|\le 1$. In particular, for each $t$, by identifying the disk bounded by $\varphi_t(S^1)$ with the disk $|z|\le 1$, we can define the cross-ratio of points $\varphi_t(0)$, $\varphi_t(\frac{1}{4})$, $\varphi_t(\frac{1}{2})$, $\varphi_t(\frac{3}{4})$ as points lying in $|z|\le 1$ . I am pretty sure that such a cross-ratio is a continuous function of $t$.

Question. Does this statement follow from some standard result? How to prove it?