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Animate a cube turning via mathematics (Java)

For a school project i have made a program that can solve a rubiks cube (you know, that cube with all the colors). But now my teachers asked if i could do some research and try out 3d animation for one of the sides. But they want to see the mathematical way to do it. I have found a way to move the corners with the use of polar coordinates. But i do not know how i could render a cube in 3d and be able to animate it.
So my question is: how can i render things like a cube in 3d (or are there any good libraries for it) and how could i use these polar coordinates to animate it?
And is there a good tutorial out there for java 3d rendering?
i must say that i have absolutely no experience with 3d rendering, so it might be a bit difficult. But i really would like to try it out.
thanks in advance

Depending on whether you want to take an existing implementation or if you want to build code for 3D animations from scratch, you might try this tutorial. Graphics programming in this area can be quite involved; a full answer of the question is beyond the scope of this site. However, there are a few main areas.
Usage of vector math for transformation of objects (translation, rotation). This can be done directly or via projective geometry using 4-by-4 matrices. The latter is easier for concatenation of transformations.
Backface culling to remove faces of the object which cannot be seen by the camera.
Using a projection and a camera model to transform 3D coordinates to 2D coordinates.
Using a rasterizer to render the 3D vector information to the screen. Here Bresenham's algorithm might be a good start.

I would suggest you look at one of the 3D libraries. One that I've used a lot and found to be excellent is JMonkeyEngine (JME) which is designed for games but would work well for your needs. It also has an excellent tutorial that takes you from basic to very complex. In fact the first step in the tutorial is a revolving coloured cube!
JME takes a huge amount of the work out of 3D modelling. You build a scene in code with materials, lights etc. and JME does all the work to render it. You can even build your models in a tool such as Blender so you don't even need to do the modelling in code. But I will warn you that using modelling tools is definitely not for the faint hearted.

Square BoundingBox with OpenGL JOGL Java

I'm trying to make a project in OpenGL using JOGL.
If you see my image http://imgur.com/DDHoXEz, I have 4 viewports with different projections but all Teapots are out of "scale", and I want to make something like a bounding box, a square with side 1, that contains all objects on the viewports, to make a scale out of the square.
Any tips?

Unless you're going to use the base teapot model for programs (which you shouldn't), I don't think this is something to spend your time on. When you get into actually using your own models, you will have direct control over the scale.
I would recommend at this point learning about different drawing methods in OpenGL (e.g., GL_TRIANGLE_FAN, GL_LINE_LOOP). Then move on to learning about vertex arrays and maybe write an OBJ importer. I can point you in the right direction if you'd like.
Here is a good place to get started on different drawing techniques.
Happy coding!

3D rendering in Java - Where should I start?

I'm thinking of making a 3D renderer, I know all about the XYZ dimensions, and am an intermediate Java programmer. But what should I do for a basic renderer? It has to be really basic, what should I render? A cube, a pyramid? A sphere?
Any answer would be much appreciated. Thank you.

My favorite first question for 3D rendering was that how to give feeling of depth with just two, perpendicular and angled from perspective, lines. Solve this and I am sure you will end up building your own 3D renderer.

I realized this is an old question but for anyone wondering I suggest you check out this page which goes through the concepts:
stuff
Then check out this tutorial on creating a 3D rotating cube. It's really easy to convert python code to a language of choice.
stuff
Theres also a javascript version here:
stuff
Then check this out, it explains creating a 3D engine:
http://blogs.msdn.com/b/davrous/archive/2013/06/13/tutorial-series-learning-how-to-write-a-3d-soft-engine-from-scratch-in-c-typescript-or-javascript.aspx

What should I use to display game graphics?

I have a system in place for a game yet I don't know what I should use to display it. I am making a vertical shooter game and I have written the methods for all the classes controlling enemies and players, but I have not idea how to efficiently display the game. I was thinking a Canvas, that would repaint every frame, but is that really the most efficient method?
Important details:
ideal framerate: 25fps
It is a 2d game
There are anywhere between 25-100 objects on the screen at any one time, all of which are moving
All objects being displayed are images, all in PNG format
The window is 640px by 480px
Right now all the images are loaded as BufferedImage, although I could easily change this
7. I need a coordinate plane. This is the only fundamental part that cannot be changed without completely restructuring my code.
Most importantly the way I have everything set up, every frame all of the objects move and interact in a coordinate plane I devised (deals with collision detection and movement, no graphical component), then everything should get painted to the screen, simply by going through the ArrayLists which keep track of all moving objects and painting them one by one.

If Swing is acceptable, JPanel is double-buffered by default, and a javax.swing.Timer with a period of 40 ms will give you ~25 Hz updates. This example shows the basic approach, while this example shows a number of images in motion.
I need a coordinate plane.
It's not unusual to have the the model and view use different coordinates; all that's needed are functions to map one system to the other. In this game, the view relies on four methods that map tiles to pixels and vice-versa. The same approach is outlined here for map tiles.

You have a number of options available to you:
Firstly, you could use one of the existing Java game frameworks:
JMonkeyEngine (http://jmonkeyengine.com/)
Slick (http://slick.cokeandcode.com/index.php)
(Slick is aimed at 2D graphics, while JMonkey is aimed at 3D and uses OpenGL - While I have looked into their use, I've not actually used them myself)
Alternatively you can code everything yourself. From the sounds of things this is your first (graphical) game, so you may want to read up on a technique known as double buffering, whereby you write each frame off-screen and the just paint the whole thing to screen, as this can lead to smoother animation.
To get you into games development a bit more, I would highly recommend reading this site, Killer Game Programming in Java by Dr Andrew Davison, as he gives some good pointers, and also provides a good progressive learning path for new game developers, and moving them into 2D and then 3D development.
HTH

The answer really depends on whether this game is 2D or 3D.
If your game is 2D, an easy way to do what you want is to use Java's own 2D Graphics API. A good tutorial to start you off (at least in my opinion) can be found at The Java Tutorials. In my experience I have found that Java's graphics API is easy to learn, and is more efficient than one might expect. The basic technique is to have your game code keep track of the positions of all your objects, then translate those coordinates into screen coordinates and display appropriate images at those locations. I made a (very, very simple) game in Java once, and this is the method I used.
If your game is in 3D, OpenGL is definetly the way to go. I have only limited experience with OpenGL, so I'm not sure how easy the Java bindings are to work with. In general, 3D programming is a massive topic, so if this is a first-time project or you aren't prepared for a major time investment, I would attempt to find a good game framework to use or make the game 2D. If you are interested in OpenGL, or 3D programming in general, a quick Google turned up the JOGL project. I would recommend investigating JOGL as a way to access the OpenGL API from within Java code, but for actually learning OpenGL I recommend The OpenGL SuperBible(5th edition), commonly known as "The Blue Book".
The code examples are all in C++, but for the OpenGL functions it could possibly be just a matter of using a wrapper library. For example:
glDrawElements(...);
May become:
JavaGLWrapperObject.glDrawElements(...);
Unfortunately I can't give concrete examples because I haven't used OpenGL with a Java program, but the above example very coarsely approximates how OpenGL ES is used on the Android platform.
As far as performance... Java's API comes with a non-trivial ammount of overhead, but I could see it doing alright for your purposes. You may have to put a little more effort into making your algorithms efficient, and your game may not run as well on less-capable hardware. If you do decide to go the OpenGL route, it will almost certainly be much faster (again, depending on how efficient your algorithms are), but is correspondingly much harder to learn and get started with. Certainly doable, but it will be a challenge.

Canvas will quickly be too slow for 25+ fps targeted.
the number of object is irrelevant, what is is how complex they are.
if it is 100 images will take nothing compared to 1 3D model of Avatar for example.
in Java you can use either Opengl or Java3D
I would tend to go with Opengl as a personel choise

Do I need to use OpenGL in my 2D Java sim game?

I have recently been developing a sim game in java, as many of my questions show, and it's come a long way! Graphically, it is somewhat lacking, and at current I'm using shapes for items, people, rooms etc.
I was looking at other similar projects, and noticed that one was using OpenGL. I have been reading CokeAndCode "space invaders 103 - refactoring and OpenGL", and I still don't know if it is something I need in my game or not.
Here is a video that I used for a previous question, but it still shows basically what I have so far (http://www.screenjelly.com/watch/Bd7d7pObyFo), although I have done a lot of refactoring and re coding over the last few weeks, but graphically, nothing has changed. As I understand it, I'm currently using java2D to do this.
I really don't know how I should continue graphically with this game. This is my first project I have done outside of learning for my own enjoyment, and so I'm still rather new to this.
Thanks in advance!

This could turn into a long subjective conversation quickly but I want to state the fact that you don't need openGL. Your application doesn't need to be the best looking one out there. Nethack still attracts people and between you and me : the graphics sucks.
Do you want to learn about 2D and 3D graphics and textures and so on ? Then try stuff... best way to learn. Then you can add this to your resume.
There are a lot of tutorial to do your first steps in OpenGL or some other graphic library. You can even try Qt Jambi. But once you start, you're in for a ride ;)

According to this, it's still largely possible to implement a working graphical game in Java2D with decent performance, as long as you make sure not to do a certain number of bad things.
It might help you to check and see how your approach is, compared to this.
I have no idea how far along Java has come with OpenGL support, so were I in your place I would probably just see how much refactoring of the Java2D code I could do, and figure out what to do from there.
Hope this helps, at least a bit.

One good side of using opengl (even for 2D graphics) is that you will get (if properly done) hardware graphics acceleration for your scenes.
Also, you can use orthogonal projection and keep one of the axis (Z) as zero to do your 2D graphics. It will be easy if you want to add a 3D effect (like lightning or something) if you use OpenGL.
However it all depends on how much you want to improve the graphics, since adding OpenGL might make things a little more complicated than plain Java2D.

How to continue graphically: use textures, more interesting animations (implement a simple physics engine and/or collision detection, that always looks nice).
When you go 3d (for example using opengl) you can add lighting effects to the equation, that usually does a lot to the aesthetics of the whole. For example if the moving circle was a sphere with a nice texture (marble or something), then you could make it really role over the board, and you could specify diffuse and specular reflection values to vary the lighting like for example here.
Camera movement can also beef up the graphics.