Here's a case where "some desktop applications aren't meant to live online".
You could certainly do this with javascript (a lot of javascript). Like you point out however is the problem of pushing the data for the drawing actions back to the server.
1. You could look into remote scripting which would allow the client side of the application to push/pull data to/from the server without the need to postbacks. This has some limitations however.
2. Alternatively, you could maintain all the drawing information on the client side and provide a means to save the data that can be done just once with a button click.
If you need to get the draw data to your server, you could always go with a lightweight desktop application that the user would download off the website which talks to the server thru web services. This of course would require the user to have the .NET framework installed. But like I said earlier, some types of programs just aren't well suited for the web.
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Peter