Following is the photo taken from page 112 of the book An Invitation to Algebraic Geometry.
The author is blowing up $\mathbb{V}(x^2+y^2+z^2)$ at the origin. My confusion is, when defining $$ V=\pi^{-1}(\mathbb{V}(x^2+y^2-z^2))\cap(\mathbb{A}\times\mathbb{A}_z^2) $$
Why aren't we restrict $z=1$? And where does the isomorphism $$ \{((x,y,z),[x:y:z]) \;|\; x^2+y^2=z^2\}\cong \{(x,y,z,u,v) \;|\; x=uz,y=vz,u^2+v^2=1\} $$ come from?
You know that $$ \{((x,y,z),[x:y:z])\}\cong \mathbb{A}^3 \times \mathbb{P}^2 $$ and $\mathbb{P}^2$ is covered by three affine space $\mathbb{A}_x^2$, $\mathbb{A}_y^2$, $\mathbb{A}_z^2$.
When you intersect the preimage with $\mathbb{A}_z^2$, $\{([x:y:z])\}$ becomes $\{((\frac{x}{z},\frac{y}{z},1))\}$.
Then let $u=\frac{x}{z},v=\frac{y}{z}$, you'll get the result.