In the context of a game, I want to draw gears. The most common curves available on the platforms I'm using are third degree Bézier curves.
Is there an exact representation of the involute gear using only Bézier curves ?
No. From this paper that discusses approximating an involute curve using a 4th degree Bezier:
While researching gear profile approximations, I came across this SE.
I wanted to add that I found this site: http://www.arc.id.au/GearDrawing.html which describes the usage of cubic (3rd degree) bezier curves about halfway down.
I know this is an old post, just thought I'd update some more relevant information because the answer was given for 4th order only. That paper is referenced heavily within the site.