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Rendering a triangular face in 3d space

Let's say I have a triangular face in 3d space, and I have the 3d coordinates of each vertex of this triangle, and would also have other information about the triangle(angles, lengths of sides, etc.). In Java, if I have the viewing screen and its information, how can I draw that plane, without using libraries like LWJGL, to that image, assuming I can properly project, accounting for perspective, any 3d point to that 2d image.
Would the best course of action just be to run a loop that draws each point on the plain to a point on the image(i.e. setting the corresponding pixel), which will most likely set the same pixel multiple times? If I'd do this, what would be the best way to identify each point in an oblique triangle, or a triangle that doesn't line up nicely with the axes?
tl;dr: I have a triangular face in 3d space, a "camera" looking at the face, and an image in which I can set each pixel. Using no GL libraries, what's the best way to project and draw that face onto the image?

Projection :
won't detail as you seems to know it
Drawing a line
you can look at Bresenham algorithm if you wanna start with the basics
(hardwared in recent graphics card)
Filling
you can fill between left and right borders of the triangle while you use Bresenham on both (you could use a floodfill algorithm starting ... i don't know, maybe at the projection of the center of the triangle)

Your best bet is to check out the g.fillPolygon() function for Java. It allows you to draw polygons with as many sides as possible and theres also g.drawPolygon() if you don't want it solid. Then you can just do some simple maths for the points. Such as each point is basically it's x and y except if the polygon is further away the points move closer to the center of the polygon and if the polygon is closer they move further away from the center of the polygon.
A second idea could be using some sort of array to store pixels and then researching line drawing algorithms and drawing lines then putting all the line data in another array and using some sort of flood-fill. Then whilst it's in that array you could try and do some weird stuff to the pixels if you wanted textures or something.

How to select a figure to move, rotate and scale with Java2D

I have problems that are not as selecting a figure and that figure selected apply a rotation as many figures as I do not do the following as the first two points and I have but not the third:
The program should allow multiple draw figures in the plane.
Each time you insert a figure in the plane should ask the dimensions of the figure and the coordinates in the insertion.
Once inserted the figure the program should allow the selected shape transformations, rotations, translations and scaling.

You've not provided enough details about your problem, but, you can take a look at the following that deal with rotation
AffineTransform.rotate() - how do I xlate, rotate, and scale at the same time?
How to mirror an image with Java.awt
Change the angle/position of a drawing with a algorithm in Java
The following deal with resizing
How do I resize images inside an application when the application window is resized?
Java: maintaining aspect ratio of JPanel background image
I also suggest you take a look through
Graphics2D
Performing Custom Painting
Using Mouse Listeners

How can I create rounded diagonal lines in Java using the ACM library?

I am trying to draw rounded diagonal lines in Java using the ACM library.
A very complicated method would involve drawing a diagonal line, scaling it up to increase its width, drawing an arc on the top of that line with respect to the angles it is rounded.

From the picture you linked to, it looks like you want lines with round end-caps, sometimes called line-joins. I don't know the ACM library too well, but looking at the docs for it, it seems like you could achieve that look by using a GPen with an image that is just a filled-in circle. You could call the setLocation(x,y) and then drawLine(dx, dy) methods on it.

I've never used ACM but looking at the javadoc you might want to consider using shapes rather then lines. So for example, you can draw a rectangle of width using GPolygon then draw circles on the ends using GOval
So something like: draw a polygon around the points [100,0],[0,100],[10,110],[110,10], then draw two circles of size 10 at 0,110 and 110,0. If those shapes are all filled with the same color, they should look like one solid.

Generating Very Large Images at Runtime with OpenGL and libgdx

I am generating very large hex grids (up to 120k total hexes at 32px wide hexes results in over 12k wide images) and I'm trying to find an efficient way to bind these to OpenGL textures in libgdx. I was thinking of using multiple FBOs and breaking the grid up as necessary into tiles, but I'm not sure how to ensure continuity between the FBOs. I can't start with one massive FBO, because that is backed up by a texture so it would fail from trying to load it to video memory. I can't use a standard bitmap on the heap because I need the drawing functionality of an OpenGL surface.
So what I was thinking was I would need to overdraw on the FBOs and somehow pick up on the next FBO exactly where the previous left off. However I'm not sure how to go about this. I'm drawing the hex grid with a series of hexagonal meshes, FYI.
Of course, there's probably some other much simpler and more efficient way to do this that I'm not even thinking of, which is why I pose this question to you fine people!

You have to draw it in pieces. You need to be able to draw your hex grid from an arbitrary position. This means being able to compute which hexes to draw based on a rectangle overlaid over the map. This isn't a hard problem, and I wouldn't worry too much about drawing extra stuff off-screen. You should master this ability to view the hexmap from any position before moving on.
Once you've mastered that, it's really simple.
Draw the top-left corner and store the pixel data. Then move the area you're drawing over exactly one image width. Draw and store that. Move the area over one image width. Draw and store it. Keep doing that until you've covered the entire width.
Move down one image height and repeat the process. Once you've run out of width and height, you're done. Save your mega-huge image.
You don't need FBOs for this. You could draw it to the screen if you wanted. Though if you want maximum performance, I would suggest using FBOs, double buffering them, and using glReadPixels though a pixel buffer object. That should cut down a lot on latency.

Using a canvas much larger than the screen

I'm trying to draw a 2D contour plot of some data on Android and I'm wondering what the best approach would be to draw those. The whole datasets can be relatively large (2k * 2k points) and zooming and moving inside the plot should be very fast. Most of the time only a small part of the data will be drawn as the user has zoomed in on the data.
My idea now would be to draw the whole plot onto a large canvas, but clip it to the portion visible on the screen, so that only that part would be really drawn in the end. I find the 2D drawing API of Android somewhat confusing and I'm not sure if this is really a feasible approach and how I would then go about executing it.
So my questions are:
Is it a good idea to draw onto a canvas much larger than the screen and use clipping to display only the relevant part?
How would I create a larger canvas and how would I select which parts should be drawn?

You should start the other way around. Instead of creating a huge canvas you should detect what part of your plot you need to draw and draw only that.
So basically you need some navigation/scrolling and you need to keep the offset from the starting point in memory to calculate where you are. Using the offset you can easily zoom in and out because you just need to scale the plot to the screen.

Is it a good idea to draw onto a
canvas much larger than the screen and
use clipping to display only the
relevant part?
A better question might be, do you have any other options. Some might argue that this is a bad idea since your going to keep memory in use when it isn't relevant to whats happening on the UI. However, from my experiences with the Canvas, I think you'll find this should work out just fine. Now, if you are trying to keep "5 square miles" of canvas in memory your definitely going to have to find a better way to manage it.
How would I create a larger canvas and
how would I select which parts should
be drawn?
I would expect that you will be creating your own "scrolling" method when the user touches the screen via overriding the onTouchEvent method. Basically your going to need to keep track of a starting point X and Y and just track that value as you move the Canvas on screen. In order to move the Canvas there are a number of built in's like translate and scale that you can use to both move the Canvas in X and Y as well as scale it when the user zooms in or out.

I don't think that is a good idea to draw your 2D contour plot on a big bitmap because you need a vector type graphics to zoom in and out in order to keep it sharp. Only pictures are good to scale down but graphs will lose thin lines or come out deformed when scaled down in bitmaps.
The proper way is to do it all mathematically and to calculate which part of the graph should be drawn for required position and zoom. Using anti_alias paint for lines and text, the graph would always come out sharp and good...
When the user zooms out, some items should not be drawn as they could not fit into the screen or would clutter it. So the graph would be always optimised for the zoom level...