Let $X \subset \mathbb{C}^n$ be a complex complete intersection surface with only ordinary double point singularities. Let $o$ be such an ordinary double point.

Let $\tilde{X}$ be the strict transform of $X$ with respect to the blow up of $\mathbb{C}^n$ at $o$. My question is, how does $\tilde{X}$ look like?

I know that locally at $o$, the strict transform should look like 2 copies of the affine plane, and should look like $X$ far away from $o$. (This can be formalized by taking a small neighborhood of $o$). But how do these two parts "glue"?

Let $X \subset \mathbb{C}^n$ be a complex complete intersection surface with only ordinary double point singularities. Let $o$ be such an ordinary double point.

Let $\tilde{X}$ be the strict transform of $X$ with respect to the blow up of $\mathbb{C}^n$ at $o$. My question is, how does $\tilde{X}$ look like?

I know that locally at $o$, the strict transform should look like a a smooth rational curve with self-intersection $−2$, and should look like $X$ far away from $o$. (This can be formalized by taking a small neighborhood of $o$). But how do these two parts "glue"?