Let $(P,\scr L)$ be an incidence space (or linear space), i.e.\ any two points are on a unique line and each line contains at least 3 points. A mapping $\delta \colon P \to P$ is called a dilatation if for any line $L \in \scr L$ one has $\delta(L) \in \scr L$ and either $\delta(L) = L$ or $\delta(L) \cap L = \emptyset$. The triple $(P, \scr L, \cdot)$ is called a dilatation space if in addition there is a binary operation $\cdot\colon P \times P \to P$ such that for all $a \in P$ the left and right multiplications $a\sb l\colon x \mapsto ax$ and $a\sb r\colon x \mapsto xa$ are dilatations of $(P,\scr L)$. The authors show that for every dilatation space the algebraic structure $(P,\cdot)$ is a quasigroup which is fibered by the lines $F \in \scr L$ satisfying $F \cdot F \subseteq F$. Furthermore, they investigate dilatation spaces which are embedded in projective spaces.