  In the maximum independent set of convex polygons problem, we are given a set
of $n$ convex polygons in the plane with the objective of selecting a maximum
cardinality subset of non-overlapping polygons. Here we study a special case of
the problem where the edges of the polygons can take at most $d$ fixed
directions. We present an $8d/3$-approximation algorithm for this problem
running in time $O((nd)^{O(d4^d)})$. The previous-best polynomial-time
approximation (for constant $d$) was a classical $n^\varepsilon$ approximation
by Fox and Pach [SODA'11] that has recently been improved to a
$OPT^{\varepsilon}$-approximation algorithm by Cslovjecsek, Pilipczuk and
W\k{e}grzycki [SODA '24], which also extends to an arbitrary set of convex
polygons. Our result builds on, and generalizes the recent constant factor
approximation algorithms for the maximum independent set of axis-parallel
rectangles problem (which is a special case of our problem with $d=2$) by
Mitchell [FOCS'21] and G\'{a}lvez, Khan, Mari, M\"{o}mke, Reddy, and Wiese
[SODA'22].
