Consider a curve $\Gamma$ in a domain $D$ in the plane $\boldsymbol R^2$. Thinking of $D$ as a piece of paper, one can make a curved folding $P$ in the Euclidean space $\boldsymbol R^3$. The singular set $C$ of $P$ as a space curve is called the crease of $P$ and the initially given plane curve $\Gamma$ is called the crease pattern of $P$. In this paper, we show that in general there are four distinct non-congruent curved foldings with a given pair consisting of a crease and crease pattern. Two of these possibilities were already known, but it seems that the other two possibilities (i.e. four possibilities in total) are presented here for the first time.

中文翻译：

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具有常见折痕和折痕图案的弯曲折叠

考虑平面 $\boldsymbol R^2$ 中域 $D$ 中的曲线 $\Gamma$。将 $D$ 视为一张纸，可以在欧几里得空间 $\boldsymbol R^3$ 中做出弯曲折叠 $P$。$P$ 的奇异集合$C$ 作为空间曲线称为$P$ 的折痕，初始给定的平面曲线$\Gamma$ 称为$P$ 的折痕图案。在本文中，我们表明通常有四个不同的非全等弯曲折叠，其中给定对由折痕和折痕图案组成。其中两种可能性已经为人所知，但另外两种可能性（即总共四种可能性）似乎是第一次在这里呈现。