Abstract--This paper presents a robot control algorithm suitable for safe reactive navigation tasks in cluttered environments. The proposed approach consists of transforming the robot workspace into the ball world, an artificial representation where all obstacle regions are closed balls. Starting from a polyhedral representation of obstacles in the environment, obtained using exteroceptive sensor readings, a computationally efficient mapping to ball-shaped obstacles is constructed using quasiconformal mappings and Möbius transformations. The geometry of the ball world is amenable to provably safe navigation tasks achieved via control barrier functions employed to ensure collision-free robot motions with guarantees both on safety and on the absence of deadlocks. AFETY of dynamical systems is receiving increasingly more attention thanks to recently developed theoretical approach, as opposed to planning-like strategies (see, e.g., and computational tools that allow us to formulate several [16] for a recent work trying to unify reactive and predictive safety-critical controllers as convex optimization control techniques).