  In this paper we introduce a model of spatial network growth in which nodes
are placed at randomly selected locations on a unit square in $\mathbb{R}^2$,
forming new connections to old nodes subject to the constraint that edges do
not cross. The resulting network has a power law degree distribution, high
clustering and the small world property. We argue that these characteristics
are a consequence of the two defining features of the network formation
procedure; growth and planarity conservation. We demonstrate that the model can
be understood as a variant of random Apollonian growth and further propose a
one parameter family of models with the Random Apollonian Network and the
Deterministic Apollonian Network as extreme cases and our model as a midpoint
between them. We then relax the planarity constraint by allowing edge crossings
with some probability and find a smooth crossover from power law to exponential
degree distributions when this probability is increased.
