Proper gluLookAt for gluCylinder

I'm trying to draw a cylinder in a specific direction with gluCylinder. To specify the direction I use gluLookAt, however, as so many before me, I am not sure about the "up" vector and thus can't get the cylinder to point to the correct direction.

I've read from another SO answer that

The intuition behind the "up" vector in gluLookAt is simple: Look at anything. Now tilt your head 90 degrees. Where you are hasn't changed, the direction you're looking at hasn't changed, but the image in your retina clearly has. What's the difference? Where the top of your head is pointing to. That's the up vector.

It is a simple explanation but in the case of my cylinder I feel like the up vector is totally unimportant. Since a cylinder can be rotated around its axis and still look the same, a different up vector wouldn't change anything. So there should be infinitely many valid up vectors for my problem: all orthogonals to the vector from start point to end point.

So this is what I do:

1. I have the world coordinates of where the start-point and end-point of the cylinder should be, A_world and B_world.

2. I project them to viewport coordinates A_vp and B_vp with gluProject:

   GLdouble A_vp[3], B_vp[3], up[3], model[16], projection[16];
   GLint gl_viewport[4];
   glGetDoublev(GL_MODELVIEW_MATRIX, &model[0]);
   glGetDoublev(GL_PROJECTION_MATRIX, &projection[0]);
   glGetIntegerv(GL_VIEWPORT, gl_viewport);
   
   gluProject(A_world[0], A_world[1], A_world[2], &model[0], &projection[0], &gl_viewport[0], &A_vp[0], &A_vp[1], &A_vp[2]);
   
   gluProject(B_world[0], B_world[1], B_world[2], &model[0], &projection[0], &gl_viewport[0], &B_vp[0], &B_vp[1], &B_vp[2]);
   

3. I call glOrtho to reset the camera to its default position: Negative z into picture, x to the right, y up:

   glMatrixMode(GL_PROJECTION);
   glLoadIdentity();
   glOrtho(0, vp_edgelen, vp_edgelen, 0, 25, -25);
   glMatrixMode(GL_MODELVIEW);
   glLoadIdentity();
   

4. I translate to coordinate A_vp, calculate the up vector as the normal to the vector A_vp — B_vp and specify the view with gluLookAt:

   glTranslatef(A_vp[0], gl_viewport[2] - A_vp[1], A_vp[2]);
   glMatrixMode(GL_MODELVIEW);
   
   GLdouble[] up = {A_vp[1] * B_vp[2] - A_vp[2] * B_vp[1], 
                    A_vp[2] * B_vp[0] - A_vp[0] * B_vp[2],
                    A_vp[0] * B_vp[1] - A_vp[1] * B_vp[0]};
   
   gluLookAt(0, 0, 0,
             B_vp[0], gl_viewport[2] - B_vp[1], B_vp[2],
             up[0], up[1], up[2]);
   

5. I draw the cylinder with gluCylinder:

   GLUquadricObj *gluCylObj = gluNewQuadric();
   gluQuadricNormals(gluCylObj, GLU_SMOOTH);
   gluQuadricOrientation(gluCylObj, GLU_OUTSIDE);
   gluCylinder(gluCylObj, 10, 10, 50, 10, 10);
   

Here is the unexpected result:

Since the cylinder starts at the correct position and since I was able to draw a circle at position B_vp, the only thing that must be wrong is the "up" vector in gluLookAt, right?

gluLookAt() is not necessary to achieve the proper perspective. It is enough to rotate the current z-vector to point to the direction the cylinder should point.