It looks like native support for Point Sprites was planned, but never got implemented in Java3D.
Short of adding it to the source code and compiling my own version of Java3D, is there any workaround that would let me create a Geometry derived class that uses Point Sprites to render?
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Well, if you think about what exactly a point is when rendered, it's really just a small circle. I'm not super familiar with Java3D, but I'm sure there's either a class that can allow you to render circles, and if not it should be straight-forward to write something that can render a circle. You'll just have to be careful to ensure that the circle is always facing the camera completely to appear as a point.
The answer I came up with, unfortunately, is no. As of today there seems to be no functionality to allow point sprites and the addition of point sprites has been dropped.