  In human spatial awareness, 3-D projective geometry structures information
integration and action planning through perspective taking within an internal
representation space. The way different perspectives are related and transform
a world model defines a specific perception and imagination scheme. In
mathematics, such collection of transformations corresponds to a 'group', whose
'actions' characterize the geometry of a space. Imbuing world models with a
group structure may capture different agents' spatial awareness and affordance
schemes. We used group action as a special class of policies for
perspective-dependent control. We explored how such geometric structure impacts
agents' behavior, comparing how the Euclidean versus projective groups act on
epistemic value in active inference, drive curiosity, and exploration
behaviors. We formally demonstrate and simulate how the groups induce distinct
behaviors in a simple search task. The projective group's nonlinear
magnification of information transformed epistemic value according to the
choice of frame, generating behaviors of approach toward an object of interest.
The projective group structure within the agent's world model contains the
Projective Consciousness Model, which is know to capture key features of
consciousness. On the other hand, the Euclidean group had no effect on
epistemic value : no action was better than the initial idle state. In
structuring a priori an agent's internal representation, we show how geometry
can play a key role in information integration and action planning.
