Polygon Clipping using Vatti's Algorithm explained

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After having read this paper by Vatti about his algorithm (https://dl.acm.org/doi/pdf/10.1145/129902.129906), I still don't get how exactly it works.

Could anyone please help me understand how Vatti's Algorithm actually works?

Here are two Triangles I just made up:

enter image description here

S: [[5, 1], [8, 1.5], [6, 7]]

C: [[1, 5], [12, 6], [6, 10]]

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