c++ - How does gluLookAt work?

From my understanding,

gluLookAt(
        eye_x, eye_y, eye_z,
        center_x, center_y, center_z,   
        up_x, up_y, up_z
    );

is equivalent to:

glRotatef(B, 0.0, 0.0, 1.0);
glRotatef(A, wx, wy, wz);
glTranslatef(-eye_x, -eye_y, -eye_z);

But when I print out the ModelView matrix, the call to glTranslatef() doesn't seem to work properly. Here is the code snippet:

#include 
#include 
#include 

#include 

#include 
#include 

using namespace std;

static const int Rx = 0;
static const int Ry = 1;
static const int Rz = 2;

static const int Ux = 4;

static const int Uy = 5;
static const int Uz = 6;

static const int Ax = 8;
static const int Ay = 9;
static const int Az = 10;

static const int Tx = 12;
static const int Ty = 13;
static const int Tz = 14;


void init() {
    glClearColor(0.0, 0.0, 0.0, 0.0);
    glEnable(GL_DEPTH_TEST);
    glShadeModel(GL_SMOOTH);
    glEnable(GL_LIGHTING);
    glEnable(GL_LIGHT0);
    GLfloat lmodel_ambient[] = { 0.8, 0.0, 0.0, 0.0 };
    glLightModelfv(GL_LIGHT_MODEL_AMBIENT, lmodel_ambient);
}


void displayModelviewMatrix(float MV[16]) {
    int SPACING = 12;
    cout << left;
    cout << "\tMODELVIEW MATRIX\n";
    cout << "--------------------------------------------------" << endl;
    cout << setw(SPACING) << "R" << setw(SPACING) << "U" << setw(SPACING) << "A" << setw(SPACING) << "T" << endl;   
    cout << "--------------------------------------------------" << endl;
    cout << setw(SPACING) << MV[Rx] << setw(SPACING) << MV[Ux] << setw(SPACING) << MV[Ax]  << setw(SPACING) << MV[Tx] << endl;
    cout << setw(SPACING) << MV[Ry] << setw(SPACING) << MV[Uy] << setw(SPACING) << MV[Ay]  << setw(SPACING) << MV[Ty] << endl;

    cout << setw(SPACING) << MV[Rz] << setw(SPACING) << MV[Uz] << setw(SPACING) << MV[Az] << setw(SPACING)  << MV[Tz] << endl;
    cout << setw(SPACING) << MV[3] << setw(SPACING) << MV[7] << setw(SPACING) << MV[11] << setw(SPACING) << MV[15] << endl;
    cout << "--------------------------------------------------" << endl;
    cout << endl;
}

void reshape(int w, int h) {
    float ratio = static_cast(w)/h;
    glViewport(0, 0, w, h);
    glMatrixMode(GL_PROJECTION);

    glLoadIdentity();
    gluPerspective(45.0, ratio, 1.0, 425.0);
}

void draw() {
    float m[16];
    glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);
    glMatrixMode(GL_MODELVIEW);
    glLoadIdentity();
    glGetFloatv(GL_MODELVIEW_MATRIX, m);

    gluLookAt(
        300.0f, 0.0f, 0.0f,
        0.0f, 0.0f, 0.0f,
        0.0f, 1.0f, 0.0f
    );
    glColor3f(1.0, 0.0, 0.0);
    glutSolidCube(100.0);
    glGetFloatv(GL_MODELVIEW_MATRIX, m);
    displayModelviewMatrix(m);
    glutSwapBuffers();

}


int main(int argc, char** argv) {
    glutInit(&argc, argv);
    glutInitDisplayMode(GLUT_DOUBLE | GLUT_RGB | GLUT_DEPTH);
    glutInitWindowSize(400, 400);
    glutInitWindowPosition(100, 100);
    glutCreateWindow("Demo");
    glutReshapeFunc(reshape);

    glutDisplayFunc(draw);
    init();
    glutMainLoop();
    return 0;
} 

No matter what value I use for the eye vector:
300, 0, 0 or
0, 300, 0 or
0, 0, 300

the translation vector is the same, which doesn't make any sense because the order of code is in backward order so glTranslatef should run first, then the 2 rotations. Plus, the rotation matrix, is completely independent of the translation column (in the ModelView matrix), then what would cause this weird behavior? Here is the output with the eye vector is (0.0f, 300.0f, 0.0f)

        MODELVIEW MATRIX
--------------------------------------------------
R           U           A           T
--------------------------------------------------
0           0           0           0
0           0           0           0
0           1           0           -300
0           0           0           1
--------------------------------------------------

I would expect the T column to be (0, -300, 0)! So could anyone help me explain this?

The implementation of gluLookAt from http://www.mesa3d.org

void GLAPIENTRY
gluLookAt(GLdouble eyex, GLdouble eyey, GLdouble eyez, GLdouble centerx,
      GLdouble centery, GLdouble centerz, GLdouble upx, GLdouble upy,
      GLdouble upz)
{
    float forward[3], side[3], up[3];
    GLfloat m[4][4];


    forward[0] = centerx - eyex;
    forward[1] = centery - eyey;
    forward[2] = centerz - eyez;

    up[0] = upx;
    up[1] = upy;
    up[2] = upz;

    normalize(forward);


    /* Side = forward x up */
    cross(forward, up, side);
    normalize(side);

    /* Recompute up as: up = side x forward */
    cross(side, forward, up);

    __gluMakeIdentityf(&m[0][0]);
    m[0][0] = side[0];

    m[1][0] = side[1];
    m[2][0] = side[2];

    m[0][1] = up[0];
    m[1][1] = up[1];
    m[2][1] = up[2];

    m[0][2] = -forward[0];
    m[1][2] = -forward[1];
    m[2][2] = -forward[2];


    glMultMatrixf(&m[0][0]);
    glTranslated(-eyex, -eyey, -eyez);
}

Answer

gluLookAt will rotate and translate the world in a way, that the camera will be located at {0, 0, 0} and looks towards the negative z-axis. This is the camera setup used in OpenGL. The camera actually never moves, the world does. Sound confusing? Hell yeah, let me try to explain :)

Lets take this example:

eye    :  {300, 0, 0}
lookat :  {  0, 0, 0}
up     :  {  0, 1, 0}


MODELVIEW MATRIX
--------------------------------------------------
R           U           A           T           
--------------------------------------------------
0           0           -1          0           
0           1           0           0           
1           0           0           -300        
0           0           0           1           
--------------------------------------------------

First we need to analyze the rotational part of the matrix: R, U and A. As you can see the right vector (R) is not in the x-axis anymore {1, 0, 0}, it is in the z-axis {0, 0, 1}. That means it is rotated by 90 degrees around the y-axis. The same happens to the eye-position. Rotating {-300, 0, 0} by 90 degrees around the y-axis lets it end up at {0, 0, -300}, voila, there it is.

It is -300 and not 300 because the world is moved and not the camera, so the world is moved in the opposite direction, still 300 units away from the camera at {0, 0, 0}. And again, it is moved towards the negative z-axis, because that is where the OpenGL camera looks to, as mentioned above.

---

Note: there is an an anomaly in your example, the normalized vector from eye-position to look-at-point must not be the same as the up-vector, so:

eye    : {0, 300, 0}
lookat : {0,   0, 0}
up     : {0,   1, 0}    

normalize(eye - lookat): {0, 1, 0} -> the same as the up-vector

will in fact not work, we must pick another up-vector, for instance {1, 0, 0}