The Wolfram Programming Cloud makes it easy to embed videos into your cloud documents.
The following Cloud CDF shows how to create a package which implements a YouTube function which takes a videoid and optional settings to create this embedding. The key function to perform the embedding is EmbeddedHTML:
https://www.wolframcloud.com/objects/bb871f6e-169a-46d3-aa26-eb2026f0531d
The link above includes many example of use, but here is one typical example:
The xPrint was just for debugging (easy to toggle by removing and adding an 'x' in front of the Print statement).
The embedding is done with the EmbeddedHTML function, which let's you insert HTML into your cloud computable document.
Arnoud, thanks for a very nice YouTube embed solution. I studied it a bit and was a little confused about the title in the linked document. It said "youtube videos into Cloud CDF". I think its embedding HTML Iframe tag and not Cloud CDF. You also used xPrint[iframe] at the end of your package code. Could you elaborate a bit on why it is used and where it is defined.
