We present a continuous model for structural brain connectivity based on the Poisson point process. The model treats each streamline curve in a tractography as an observed event in connectome space, here the product space of the gray matter/white matter interfaces. We approximate the model parameter via kernel density estimation. To deal with the heavy computational burden, we develop a fast parameter estimation method by pre-computing associated Legendre products of the data, leveraging properties of the spherical heat kernel. We show how our approach can be used to assess the quality of cortical parcellations with respect to connectivity. We further present empirical results that suggest that “discrete” connectomes derived from our model have substantially higher test-retest reliability compared to standard methods. In this, the expanded form of this paper for journal publication, we also explore parcellation free analysis techniques that avoid the use of explicit partitions of the cortical surface altogether. We provide an analysis of sex effects on our proposed continuous representation, demonstrating the utility of this approach.