What does an elliptic curve look like near the point at infinity (the identity)?

The appearance of an elliptic curve from the point of view of the affine $(x,y)$ plane is familiar to us, but leaves us wondering what the curve might look like near the point at infinity (i.e. the identity element $\mathcal{O}$). This is not merely of visual interest, as it allows one to see directly that e.g. a line that is vertical in the $(x,y)$ plane has a pole of order 2 at the identity (even though it also intersects the curve at that point). The divisors of such linear functions play a crucial role in e.g. the computation of the Weil pairing.

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