How to texture a random convex quad in openGL - opengl

Question

Alright, so I started looking up tutorials on openGL for the sake of making a minecraft mod. I still don't know too much about it because I figured that I really shouldn't have to when it comes to making the small modification that I want, but this is giving me such a headache. All I want to do is be able to properly map a texture to an irregular concave quad.
Like this:
I went into openGL to figure out how to do this before I tried running code in the game. I've read that I need to do a perspective-correct transformation, and I've gotten it to work for trapezoids, but for the life of me I can't figure out how to do it if both pairs of edges aren't parallel. I've looked at this: http://www.imagemagick.org/Usage/scripts/perspective_transform, but I really don't have a clue where the "8 constants" this guy is talking about came from or if it will even help me. I've also been told to do calculations with matrices, but I've got no idea how much of that openGL does or doesn't take care of.
I've looked at several posts regarding this, and the answer I need is somewhere in those, but I can't make heads or tails of 'em. I can never find a post that tells me what arguments I'm supposed to put in the glTexCoord4f() method in order to have the perspective-correct transform.
If you're thinking of the "Getting to know the Q coordinate" page as a solution to my problem, I'm afraid I've already looked at it, and it has failed me.
Is there something I'm missing? I feel like this should be a lot easier. I find it hard to believe that openGL, with all its bells and whistles, would have nothing for making something other than a rectangle.
So, I hope I haven't pissed you off too much with my cluelessness, and thanks in advance.
EDIT: I think I need to make clear that I know openGL does perspective transform for you when your view of the quad is not orthogonal. I'd know to just change the z coordinates or my fov. I'm looking to smoothly texture non-rectangular quadrilateral, not put a rectangular shape in a certain fov.

Answer 1

OpenGL will do a perspective correct transform for you. I believe you're simply facing the issue of quad vs triangle interpolation. The difference between affine and perspective-correct transforms are related to the geometry being in 3D, where the interpolation in image space is non-linear. Think of looking down a road: the evenly spaced lines appear more frequent in the distance. Anyway, back to triangles vs quads...
Here are some related posts:
How to do bilinear interpolation of normals over a quad?
Low polygon cone - smooth shading at the tip
https://gamedev.stackexchange.com/questions/66312/quads-vs-triangles
Applying color to single vertices in a quad in opengl
An answer to this one provides a possible solution, but it's not simple:
The usual approach to solve this, is by performing the interpolation "manually" in a fragment shader, that takes into account the target topology, in your case a quad. Or in short you have to perform barycentric interpolation not based on a triangle but on a quad. You might also want to apply perspective correction.

Answer 2

The first thing you should know is that nothing is easy with OpenGL. It's a very complex state machine with a lot of quirks and a poor interface for developers.
That said, I think you're confusing a lot of different things. To draw a textured rectangle with perspective correction, you simply draw a textured rectangle in 3D space after setting the projection matrix appropriately.
First, you need to set up the projection you want. From your description, you need to create a perspective projection. In OpenGL, you usually have 2 main matrixes you're concerned with - projection and model-view. The projection matrix is sort of like your "camera".
How you do the above depends on whether you're using Legacy OpenGL (less than version 3.0) or Core Profile (modern, 3.0 or greater) OpenGL. This page describes 2 ways to do it, depending on which you're using.
void BuildPerspProjMat(float *m, float fov, float aspect, float znear, float zfar)
{
float xymax = znear * tan(fov * PI_OVER_360);
float ymin = -xymax;
float xmin = -xymax;
float width = xymax - xmin;
float height = xymax - ymin;
float depth = zfar - znear;
float q = -(zfar + znear) / depth;
float qn = -2 * (zfar * znear) / depth;
float w = 2 * znear / width;
w = w / aspect;
float h = 2 * znear / height;
m[0] = w;
m[1] = 0;
m[2] = 0;
m[3] = 0;
m[4] = 0;
m[5] = h;
m[6] = 0;
m[7] = 0;
m[8] = 0;
m[9] = 0;
m[10] = q;
m[11] = -1;
m[12] = 0;
m[13] = 0;
m[14] = qn;
m[15] = 0;
}
and here is how to use it in an OpenGL 1 / OpenGL 2 code:
float m[16] = {0};
float fov=60.0f; // in degrees
float aspect=1.3333f;
float znear=1.0f;
float zfar=1000.0f;
BuildPerspProjMat(m, fov, aspect, znear, zfar);
glMatrixMode(GL_PROJECTION);
glLoadMatrixf(m);
// okay we can switch back to modelview mode
// for all other matrices
glMatrixMode(GL_MODELVIEW);
With a real OpenGL 3.0 code, we must use GLSL shaders and uniform variables to pass and exploit the transformation matrices:
float m[16] = {0};
float fov=60.0f; // in degrees
float aspect=1.3333f;
float znear=1.0f;
float zfar=1000.0f;
BuildPerspProjMat(m, fov, aspect, znear, zfar);
glUseProgram(shaderId);
glUniformMatrix4fv("projMat", 1, GL_FALSE, m);
RenderObject();
glUseProgram(0);
Since I've not used Minecraft, I don't know whether it gives you a projection matrix to use or if you have the other information to construct it.