We study the blow-up curve of a (blow-up) solution to the semi-linear wave equation in 1D with power nonlinearity: $u_tt - u_xx = |u|^{p-1} u$. The blow-up curve is a priori 1-Lispchitz. On this curve, we distinguish (geometrically) characteristic points and non-characteristic points. We describe the blow-up behavior in each case, following a series of papers by Frank Merle and Hatem Zaag: in particular, the set of characteristic points is discrete, the blow-up curve is corner-shaped at every characteristic point, and is $C1$ around any non-characteristic point. We also construct construct a blow-up solution with prescribed characteristic point.