Related:

1) a tutorial on the same topic by Red Blob Games: http://www.redblobgames.com/articles/visibility/

2) a public domain Javascript visibility polygon library that runs in O(n log n), which is actually used in the Nothing to Hide game: https://code.google.com/p/visibility-polygon-js/

3) a blog post by author of said library discussing the algorithm: http://byronknoll.blogspot.ca/2013/05/on-log-n.html

Shooting a ray to every vertex and then naively computing intersections takes O(n^2) since there are n rays with up to O(n) intersections each. It is interesting to read and think about O(n log n) algorithms for computing the visibility polygon.