This work proposes a representation that comprises both shape and friction, as well as the exploration strategy to gather them from an object. The representation is developed under a common probabilistic framework, particularly it uses a Gaussian Process to approximate the distribution of the friction coefficient over the surface, also represented as a Gaussian Process. The surface model is exploited to compute straight lines (geodesic flows) that guide the exploration. The exploration follows these flows by employing an impedance controller in pursuance of safety, shape accommodation and contact enforcement, while measuring the necessary data to estimate the friction coefficient. The exploratory probes consist of an RGBD camera and an Intrinsic Tactile sensor (ITs) mounted on a robotic arm. Experimental results give evidence for the effectiveness of the algorithm in the friction coefficient gathering and enrichment of the object representation.