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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.

3D Shadow implementation idea

Lets assume your eye is in the surface point P1 on an object A and there is a target object B and there is a point-light source behind object B.
Question: am i right if i look to the light source and say "i am in a shadow" if i cannot see the light because of object B ?. Then i flag that point of object A as "one of the shadow points of B on A" .
If this is true, then can we build a "shadow geometry"(black-colored) object on the surface of A then change it constantly because of motion of light,B,A, etc... in realtime ? Lets say a sphere(A) has 1000 vertices and other sphere (B)has 1000 vertices too, so does this mean 1 milion comparations? (is shadowing, O(N^2) (time) complexity?). I am not sure about the complexity becuse the changing the P1(eye) also changes the seen point of B (between P1 and light source point). What about the second-order shadows and higher (such as lights being reflecting between two objects many times) ?
I am using java-3D now but it doesnt have shadow capabilities so i think of moving to other java-compatible libraries.
Thanks.
Edit: i need to disable the "camera" when moving the camera to build that shadow. How can i do this? Does this decrease the performance badly?
New idea: java3D has built-in collision detection. I will create lines(invisible) from light to target polygon-vertex then check for a collision from another object. If collision occurs, add that vertex corrd. to the shadow list but this would work only for point-lights :( .
Anyone who supplys with a real shade library for java3d, will be much helpful.
Very small sample Geomlib shadow/raytracing in java3D would be the best
Ray-tracing example maybe?
I know this is a little hard but could have been tried by at least a hundred people.
Thanks.

Shadows is probably the most complex topic in 3D graphics programming, and there are many approaches, but the best option should be identified according to the task requirements. The algorithm you are talking about is the simplest way to implement shadows from a spot light source onto the plane. It should not be done on the CPU, as you already use GPU for 3D rendering.
Basically the approach is to render the same object twice: once from the camera view point, and once from the light source point. You will need to prepare model view matrices to convert between these two views. Once you render the object from the light point, you get the depth map, in which each point lies closest to the light source. Then, for each pixel of the normal rendering, you should convert its 3D coordinates into the previous view, and check against the corresponding depth value. This essentially gives you a way to tell which pixels are covered by shadow.
The performance impact comes from rendering the same object twice. If your task doesn't assume high scalability of shadow casting solution, then it might be a way to go.
A number of relevant questions:
How Do I Create Cheap Shadows In OpenGL?
Is there an easy way to get shadows in OpenGL?
What is the simplest method for rendering shadows on a scene in OpenGL?

Your approach can be summarised like this:
foreach (point p to be shaded) {
foreach (light) {
if (light is visible from p)
// p is lit by that light
else
// p is in shadow
}
}
The funny fact is that's how real-time shadows are done today on the GPU.
However it's not trivial for this to work efficiently. Rendering the scene is a streamlined process, triangle-by-triangle. It would be very cumbersome if for every single point (pixel, fragment) in every single triangle you'd need to consider all other triangles in other to check for ray intersection.
So how to do that efficiently? Answer: Reverse the process.
There's a lot fewer lights than pixels on the scene, usually. Let's take advantage of this fact and do some preprocessing:
// preprocess
foreach (light) {
// find all pixels p on the scene reachable from the light
}
// then render the whole scene...
foreach (point p to be shaded) {
foreach (light) {
// simply look up into what was calculated before...
if (p is visible by the light)
// p is lit
else
// p is in shadow
}
}
That seems a lot faster... But two problems remain:
how to find all pixels visible by the light?
how to make them accessible quickly for lookup during rendering?
There's the tricky part:
In order to find all points visible by a light, place a camera there and render the whole scene! Depth test will reject the invisible points.
To make this result accessible later, save it as a texture and use that texture for lookup during the actual rendering stage.
This technique is called Shadow Mapping, and the texture with pixels visible from a light is called a Shadow Map. For a more detailed explanation, see for example the Wikipedia article.

Basically yes, your approach will produce shadows. But doing it point by point is not feasible performance wise (for realtime), unless its done at the GPU. I'm not familiar with what the API's offer today, but I'm sure any recent engine will offer some shadow out of the box.
Your 'New idea' is how shadows were implemented back in the days when rendering was still done with the CPU. If the number of polygons isn't too big (or you can efficently reject entire bunches by having grouping volumes etc.) it can be done with fairly little CPU power.

3D shadow rendering on vanilla Java is never going to be efficient. You best use graphical libraries written to utilize the full capabilities range of the graphical card, such as OpenGL or DirectX. As you are using Canvas (from the screenshot you provided), you can even paint that Canvas from native code using JNI. So you could use all the technology from graphial libraries, do just a little fiddling and paint your Canvas directly from the native code. There would be very little work involved to make it work, compared to writing your own 3D engine.
Wiki link about AWT native access: http://en.wikipedia.org/wiki/Java_AWT_Native_Interface
Documentation: http://docs.oracle.com/javase/7/docs/technotes/guides/awt/AWT_Native_Interface.html

converting a 2d window point to a 3d point

i am writing a 3d modeler similar to Blender for a game i am making. Since programs like blender export very complicated file types with alot of unneeded data i wanted to write a simple editor for my game. what i cannot figure out is how to map a point from a 2d projection on the window to where i have clicked in the 3d world with the world being rotated.
If anyone knows any good tutorials on how to do this or the method any help would be appreciated. I know i could use ray tracing but that would be to complicated i think.

The two main methods of mouse picking are:
Intersection Testing
Color Picking
Intersection tests are the more popular of the two, and at the simplest level involves 'shooting' out a ray and checking if it has intersected any points. The ray can also be replaced by a polytope if one wants to achieve more sensitive picking (useful for choosing points on vertices).
Color picking involves disabling AA, blending, shadows, etc. and re-drawing the scene using solid colors for the objects. glReadPixels is then used to find the color at the point of the mouse and this color is used to determine if it clicked on an applicable object.
Ray Picking:
Mouse Ray Picking Explained
Picking, Alpha Blending, Alpha Testing, Sorting
Color Picking:
OpenGL Selection Using Unique Color IDs
Picking Tutorial

The term you are looking for is mouse picking.

The method you need is gluUnProject. You'll need window x,y and the depth.
I think, in your case, it might be a lot easier to write a simple exporter for Blender.

Java/Graphics: Plotting 3d coordinates?

I'm coding a programm, that will produce 3d coordinates for a rocket and I would like to do a rudimentally graphic output for this.
it just has to be scaleable and rotationable, so that you can change the view manually.
the postions should be connected by lines and
it ould be nice to have spheres for earth and moon ( and perhaps addtional celestial objects).
I think, there should be some ready stuff for this kind of plott already available, but I couldn't find one.
So that's why I'm here to ask you, if you know such a thing.
and if there isn't I would kike to ask you how a bignner like me should start this?
I ust coded for console apllications, because there was no need for a real graphics output.
thank you in advance for any tip! :)
Andreas

I don't think there are sand-box ready things you can use to draw customizable 3d objects in Java, if you can live without strange things you can you just a graph library able to draw 3d graphs like for example jMathTools (link).. otherwise you should go into J3D with opengl and similar things.
I don't think they exist just because doing simple things is trivial if you work with OpenGL or similar APIs..
Doing what you need with OpenGL is not complex at all, just a GL_LINE_STRIP to draw the trajectory and some primitives if you need earth, moon and so on.. rotating and scaling come implicitly moving the camera of your viewport..
Take a look at: Java3D or JOGL

I can't believe nobody has mentioned this yet, but the NASA WorldWind project seems like exactly what you need: http://worldwind.arc.nasa.gov/java/ You can extend it with JOGL if needed, or you can use some of the vast modeling objects already available to mark trajectory and location in 3D coordinates, complete with zooming/rotating and the like. Having accurate Earth layers is nice as well =)

Use JOGL! It is easy to install, and if you know Java it should be easy. There are a few things in JOGL that require you to be decent in trigonometry. This tutorial might help you get started: http://www.youtube.com/watch?v=rT02jFYrXv0

how can i do to a 2D image and spin it around it's X, Y, or Z axis as if it were a 3D image.in java?

I want to achieve the effect of a 2D image I have but a little inclined, for example a plane, I want the image can be rotated about its axis Y. .. anyone can help me with some idea of how to do ..**

Basically you need a little linear geometry/algebra, and/or a package to do them for you.
From the geometry point of view, you think of the image as if it's on a plane in space; you're looking at it as if it were back-projected on your monitor. If the picture is exactly parallel to that screen, and the same size, each point is mapped to a pixel on the screen. Otherwise you have to go through a computation that makes that mapping, which involves a trig function for the angles in the x,y,z directions between that plane and the plane of the screen. The linear algebra comes in because the easy way to handle this computation is as a series of multiplications of 4×4 matrices.
Now, you could program all that yourself, and for what you're thinking of it wouldn't be all that difficult. See any good computer graphics text, like Shirley, or Foley and van Damm.
As far as a package, there's good 3D graphics in Java. Even better, there are good tutorials:
The Java3D tutorial at Sun.
the stuff at j3d.org
a whole list of them at java3d.org

In what context ? Using a 3D API like OpenGL trough JOGL seems to me like the simplest way to achieve this. Otherwise, if the angle is variable, you'll need some form of software renderer.