How can the Neron-Tate height pairing be considered an intersection number?

Question

Given the Neron-Tate height $\hat h$ on an elliptic curve we defined the associated bilinear form: $$\langle P,Q\rangle = \frac{1}{2} \bigl( \hat h(P+Q) - \hat h(P) - \hat h(Q) \bigr) $$ My professor offhandedly mentioned that this form can be considered a sort of "intersection number", or specifically when $P=Q$ a self-intersection number. The notation certainly seems to suggest it, but I can't immediately see why. How is Neron-Tate an intersection number?