If you care about speed, then there is a linear-time algorithm specifically tuned to intersecting
two convex polygons, described in the book,
<a href="http://cs.smith.edu/~orourke/books/compgeom.html">*Computational Geometry in C*</a>,
with downloadable code.
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<img src="https://i.sstatic.net/ofx83.png" alt="Fig.7.15" />
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(I took the above snapshot from an illegal scan of the book!)