How can I attach two pieces of wood in an invisible way?
I would suggest a tenon cutter.



It will mount in a drill. You put a tenon on the end of your wood of a specific diameter, then drill a hole (a round mortise) in the wood it will be attached to. The tenon slips right in. Use a little glue, and it will stay firmly there.

As well, this will yield a very strong joint. For example, look at how the legs on Windsor style chairs fit into the seat from underneath. Such tenons are also used on the braces between the legs in such chairs, where one member is tenoned into another.