The cycle convexity of a graph, introduced by Norbert Polat in [13], is a convexity finer than the standard geodesic convexity. It arose from his investigation of special classes of partial cubes known as netlike partial cubes. In this paper, we prove that cycle convexity is associated with a transit function of arity three, which we name the cycle transit function, denoted as the \(C_3\) -function. We investigate some natural extensions of the betweenness properties of arbitrary 2-ary transit functions as they apply to the \(C_3\) -function. Using a set of first-order axioms, we also characterize the betweenness properties of its arity-two counterpart, the \(C_2\) -function, for certain special cases. We also study another natural extension of the geodesic interval function I: the \(I_3\) -function and prove that, in any connected graph, some of the natural extensions of the betweenness axioms for the \(C_3\) -function and the \(I_3\) -function are equivalent.