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HTML Rendering Engine as a Java Control

We have a client side application (Java/Swing) that we need an HTML rendering control for.
What I want to find is the most widely adopted, most heavily developed, easiest to deploy solution to get Gecko or WebKit into a Swing app (Needs to run OS X and Windows).
The limited (crappy?) JEditPane type solutions are not robust enough for our needs. We would really like to use either WebKit or Gecko.
Some libraries seems to exist that would allow this:
(QT WebKit) http://trac.webkit.org/wiki/QtWebKit
(JRex) [can not post URL because I am new]
etc.
Whats the best library to achieve this?

Doing a quick Google search tells me that there is a JWebPane in JavaFX.
Have a look here.

Really quick and dirty solution for embedding webkit in a Swing UI?

I am looking for a really quick and dirty solution for embedding webkit in a Swing application.
What I am looking for is a way to implement a panel that contains a webkit browser running some JavaScript that will be able to do one or more of the following:
Call a Java function of the containing application
Send JSON objects to the containing application
I would also like to have access to the DOM from my Java code.

JWebPane is allegedly going to do this for you, but it's vaporware for the time being. Your best bet might be to take a look at the WebKit component in Qt Jambi and use the Swing bridge to stick it in a Swing app.

How can a java library be ported to the web?

How do you take a java library that is for the desktop or an applet and make it so you can use its functions for a web page? I know not all things are meant for the web, but can you call a "regular" java class in an apache/tomcat setup server and it "work"? Do you have to so something to it to get it work with a web page?
I was interested in jgrapht for the web but it looks like all desktop or applet and I don't know that it would be possible to return its graphics to the browser, but maybe its libraries and a generated .png rendered by the browser.
Although there is debate on applets being alive or dead, I am not interested in using one at the moment, as it appears it is dead, albeit with new possible life in the latest update 6u10. I don't know that I want to invest in JavaFX either.

Johnny, the thing is that you have to decide what you mean to do. You could, for example, use JGraphT in server side code; the graph could be visualized in a bunch of different ways, like JGaph or graphviz. Then you would output the visualization in some form that works on the web and put it into your web page. In JGraph, that probably means a print interface that generates SVG or PNG.
Or you could emit the graph structure as JSON with a simple walk of the graph and interpret it on the fly in the browser using Javascript.
It's hard to answer the question generally, because it strongly depends on how you mean to use it.

Yes, it is possible to use a java class or jar file in a web appliaction, but it also depends on what you want to do with it.
For example if the library is for mathematics and you want to return results back to the users on a web based application, this is definitely possible. There are many other examples I can give but they are all the same.
As for displaying charts, I have successfully used JFreeChart in a web application using Apache Wicket (which rocks btw).
So to answer your question, you don't really need to "port" the library, you just need to use it and display it's output to a web page instead of a java desktop GUI.
Of course their might be some libraries that are specifically designed for Java GUI stuff like Swing or AWT. But for the most part there is no difference in using a Java library in a J2EE environment or a J2SE environment.

There isn't a closed-form answer to what you are suggesting: porting a Java Library to a browser's environment and using its functionality.
If you are interested in graphing things, here are a few things you can try:
Open Flash Chart - a scriptable flash app that will produce graphs
Bluff - a Javascript library that produces graphs
processing.js - a port of the excelling 'processing' library to Javascript
As for accessing Java-based functionality in the browser, there are some approaches for that. One is to use a toolset like GWT that complies java into an HTML/Javascript/Java application. Another is to use an AJAX-based technology like Direct Web Remoting. It really depends on what you are trying to accomplish.

Embedding Flash / Flex component into Java app

I'm working on some Flex spike in my company. We are basically evaluating different scenarios etc. What solution would you recommend for embedding Flex components into Java app? Flex <-> Java communication is not (yet...) an issue, just embedding swf into JFrame.

I've done it with EasyJCom. It's pretty straight forward as long as you're using one of the standard Java windowing libraries (Swing, awt). You can see an example (From the EZJCom site) here: http://www.ezjcom.com/FlashTest.java.txt
The people responsible for EasyJCom are also very responsive, and even though we ended up not using their product (Switching platforms), I'd recommend them.
I haven't used JFlashPlayer, but did take a look at the docs, it seems to be adequate as well. I think their license requires you to obfuscate your code. I thought it was a little odd, but maybe I mis-understood the license.
It gets quite a bit more complicated if you don't want to use Com, or if you need to embed in another type of window.
If you need to embed flash cross platform, you may be stuck rolling your own Jini library for the Mozilla plugin. At that point... Maybe it's worth just switching platforms if you aren't too far in. ;)

Haven't tested this, but it looks like JFlashPlayer will do the job. http://www.jpackages.com/jflashplayer/

None of these supports Linux or MacOS. So much for the cross-platform Java and Flash.

I think the best way to do something like this would be to embed something like xulrunner into your java app then have the java app interact with flash via the xulrunner interfaces.

.NET, Java to JavaScript compiler

I am interested to create a drag-and-drop layout designer using only JavaScript, HTML and CSS. The designer will allow the user to drag the page elements from one place to another (something like Blogger's layout designer) to create a site layout. But I don't want to hand code everything in JavaScript, I would prefer to write my application in .NET (preferably) or Java and rely on a compiler to compile it to JavaScript and HTML.
What are the .NET or Java to JavaScript compilers that you have used and can recommend? For Java to JavaScript I know GWT is available. What about .NET to JavaScript? Microsoft did come out with Volta, but the project seems to be no longer available.

Look no further, you already mentioned GWT pick that!
It has a very good API and many good applications have use them.
Even JavaScript frameworks like http://extjs.com/ have GWT support.
I use it for an small JavaScript calendar recently.
To be honest, I don't really like JavaScript that much. Most of the times the errors are hard to track (specially for a non JavaScript guy as me) and the workarounds included some plug-ins for the explorer just to get exactly what a compiler should do. Catch silly error early.
In the other hand I'm very familiar with the Java Programming language, and many of the libraries (if not the most important) such as java.lang and java.util have been ported to GWT.
Plus, the guy who wrote relevant parts of java.util is the same behind GWT (google Joshua Bloch.)

Check out Nikhil Khotari's Script# project. It allows you to write C# code and compiles it to JavaScript.

Script# has already been mentioned. It hasn't been updated since August 2008.
Milescript is another, but also has seen very little for 6 months.
Extsharp for the Ext library. Adds Ext support for Script#
Javascript compiler to Java (going the wrong way for you)
Java to script Eclipse plugin
My issues with Script# (a known issue) is it doesn't support jQuery yet. However it comes with a very lightweight library to tie in with the .NET framework, in Nikhil's sscorlib.js file and ssfx.core.js files. And also has support for lots of other Javascript APIs (mostly Microsoft, seeing as he is in the ASP.NET team).
I'd love to see a Script# extension for jQuery (I'm thinking about writing it if it's easy enough). As it stands, most don't provide full compilation yet but they're certainly getting there.
Update: I wrote a small extension to enable JQuery support Script# a few months ago. The project can be found here.

I'm going to second the use of GWT. I've used it for several projects and, when used in combination with a modern editor like Eclipse or IDEA, it really makes the mess manageable.
It's important to note that not only does it allow you to write in Java and have that transformed into optimized and obfuscated Javascript it also comes with a substantial subset of the core Java API. In addition to this they provide lots of additional classes for doing things like parsing and working with JSON and XML and communicating with a server via asynchronous HTTP. You can check out the docs to get an idea of what else they offer.
Another feature that might be of special interest to you for implementing drag and drop functionality is it's integration with javascript libraries like Ext and scriptaculous. Either through pre-built interfaces or via JSNI

Also for Java there is J2S.
Java2Script (J2S) Pacemaker provides
an Eclipse Java to JavaScript compiler
plugin and an implementation of
JavaScript version of Eclipse Standard
Widget Toolkit (SWT) with other common
utilities, such as java.lang.* and
java.util.*. You can convert your
SWT-base Rich Client Platform (RCP)
into Rich Internet Application (RIA)
by Java2Script Pacemaker.
This means that if you use the SWT IDE (drag and drop) you can then convert the generated code to JS + HTML.

I wouldn't hand write any Javascript for UI. This can lead to maintenance disaster. jQuery is what I am using but I still wouldn't use it to write full UI Javascript code. ExtJS is also another good option if you plan to write in Javascript. In general what I am saying here is that it's so much easier to main in Java/C# than Javascript. Check out cappuccino framework and Atlas. Never used GWT. Script# is similar to GWT but for ASP.NET framework. Also depends on the requirement, if your site is public facing then RIA isn't a good option. It's all about which extreme end you pursuit (hand written and web standard, or RAD or libraries like jQuery/ExtJS as the middle option).

Check out Axial, a .NET to JavaScript converter that works well in ASP.NET. It supports WebForms, jQuery and canvas. It's not very mature, but it's worth a look.

http://jsc.sourceforge.net/ is a C# to JavaScript, Java, Flash and PHP compiler.

JscriptSuite offers another free .NET to Javascript compiler. There is a big difference to Saltarelle (jsc, SharpKit# etc.). Developer write down and debug only C# code (or any other .NET langauge), like in GWT. Javascript will be generated für deployment only.