Central Projection and Existence of the Vector Space

Question

My definition of a central projection is the following:

Let $U_1, U_2 \subseteq \mathbb{P} (V) := \{ U \subseteq V \, \text{ subspace } | \, \text{dim}_K(U) = 1 \}$ be two projective subspaces with same dimension. A map $\varphi: U_1 \to U_2$ is called central projection, if there exists a projective subspace $Z \subseteq \mathbb{P}(V)$ such that

1. $Z \vee U_1 = Z \vee U_2 = \mathbb{P}(V)$
2. $Z \cap U_1 = Z \cap U_2 = \emptyset$
3. For all $P \in U_1$ one has that $\varphi(P) = (Z \vee \{ P \}) \cap U_2$

Notice that for $A, B \subseteq \mathbb{P}(V)$ denotes $A \vee B$ the intersection of all projective subspaces $U \subseteq \mathbb{P}(V)$ with $A \cup B \subseteq U$.

My question is: Is it possible to give a construction of $Z$? Exists a general method to show it's existence, if I would like to proof that a given $\varphi$ is a central projection?

I started with $U_1 = U_2 = \mathbb{P}(V)$. Then it is easy to show that the only possible $Z$ is the empty set itself, but this is not a nice example...

My next idea was to use the basis extension theorem to get a $Z$, but I failed generating one satisfying $1.$ and $2.$ simultaneously.

Any help would be appreciated!