Rat exploratory behavior consists of regular excursions into the environment from a preferred place termed a home base. A phase plane representation of excursions reveals a geometrical pattern that changes during exploration in both shape and size. We first show that with time and repeated exposures to the same large environment there is a gradual increase in the length of excursions; each rat has its own characteristic length of excursions; but all rats share a similar rate of excursion growth. As in our experimental setup the rats perform increasingly longer paths from one location, while locomoting back and forth along the walls of the arena, exposure is more extensive at the proximal part of the route, and less at the distal part. We consequently show that the rat's velocity pattern changes concurrently with the increase in excursion length, and in correlation with the level of exposure (familiarity) to places. The primitive velocity pattern consists of slow progression while moving away from base and fast progression while returning to it. During exposure the asymmetry in velocity is inverted. The inversion spreads across successive excursions from the home base outwards. The rate of spread of this inversion is higher than the rate of increase in excursion length, and is similar across rats. Because it spreads more rapidly than the increase in excursion length, the global shape of the excursion trajectory changes. The dynamics of excursion shape share similar properties with the dynamics of excursion length. Both might reflect the same intrinsic constraints on the amount of novelty that a rat can handle per excursion.