Editing User:Hg200

= Investigations on update_camera() =

In the second part of <math display="inline">\rightarrow</math> "axes::properties::update_camera ()" the view transformation "x_gl_mat1" and projection matrix "x_gl_mat2" are put together. The following article illustrates some of the properties of "x_gl_mat1" and "x_gl_mat2".

== OpenGL coordinate systems ==

In the Octave plotting backend, we find various OpenGL transformations. Some of the classic OpenGL transformation steps, as well as coordinate systems, are shown in the following picture: 

[[File:Octave_coordinate_systems.png|center|350px]]

== The Octave coordinate system ==

In Octave a plot scene is defined by a "view point", a "camera target" and an "up vector".

[[File:Octave_view_point_setup_to_scale.png|center|750px]]

== The role of "x_gl_mat1" ==

The following section of code composes the matrix "x_view", which is a subset of "x_gl_mat1". The matrix "x_gl_mat1" consists of multiple translations, scales and one rotation operation.

{{Code|Section of axes::properties::update_camera ()"|<syntaxhighlight lang="C" style="font-size:13px">
  // Unit length vector for direction of view "f" and up vector "UP"
  ColumnVector F (c_center), f (F), UP (c_upv);
  normalize (f);
  normalize (UP);

  // Scale "UP" vector, so that norm(f x UP) becomes 1
  if (std::abs (dot (f, UP)) > 1e-15)
    {
      double fa = 1 / sqrt (1 - f(2)*f(2));
      scale (UP, fa, fa, fa);
    }

  // Calculate the vector rejection UP onto f
  // s, f and u are used to assemble the rotation matrix l
  ColumnVector s = cross (f, UP);
  ColumnVector u = cross (s, f);
  
  // Construct a 4x4 matrix "x_view" that is a subset of "x_gl_mat1"
  // Start with identity I = [1,0,0,0; 0,1,0,0; 0,0,1,0; 0,0,0,1]
  Matrix x_view = xform_matrix ();
  // Step #7 -> #8
  scale (x_view, 1, 1, -1);
  Matrix l = xform_matrix ();
  l(0,0) = s(0); l(0,1) = s(1); l(0,2) = s(2);
  l(1,0) = u(0); l(1,1) = u(1); l(1,2) = u(2);
  l(2,0) = -f(0); l(2,1) = -f(1); l(2,2) = -f(2);
  // Step #6 -> #7 (rotate on the Z axis)
  x_view = x_view * l;
  // Step #5 -> #6
  translate (x_view, -c_eye(0), -c_eye(1), -c_eye(2));
  // Step #4 -> #5
  scale (x_view, pb(0), pb(1), pb(2));
  // Step #3 -> #4
  translate (x_view, -0.5, -0.5, -0.5);
</syntaxhighlight>}}

To visualize the matrix properties, the "x_gl_mat1" matrix is multiplied by the object coordinates. The plot box is now aligned with the Z-axis and the view point is at the origin <math display="inline">[0,0,0]</math>. The matrix transforms world coordinates into camera coordinates. The purple planes show the near and far clipping planes.

[[File:Octave_x_gl_mat1_setup.png|center|250px]]

The individual translation, scaling and rotation operations of "x_gl_mat1", are shown in the following figure:

[[File:Octave_x_gl_mat1_steps.png|center|1100px]]

== The role of "x_gl_mat2" ==

The matrix "x_gl_mat2" is composed of the sub matrices "x_viewport" and "x_projection". The purpose of these matrices is to fit the 2D image of the above transformation result into a "bounding box". The bounding box is defined as follows:

  bb(0), bb(1): Position of the "viewport"
  bb(2), bb(3): Width and height of the "viewport"

Hint: If you debug in "update_camera ()", you can print "bb":  

  (gdb) print *bb.rep.data@bb.rep.len

Compare the result with the output in the Octave prompt:

  hax = axes ();
  get (hax, "position");

In the following simplified code section the matrix "x_projection" is assembled. It is used to scale the image of the above transformation into the bounding box of the window. For this purpose, the field of view (FOV) must be calculated:

{{Code|Section of axes::properties::update_camera ()"|<syntaxhighlight lang="C" style="font-size:13px">
  if (cameraviewanglemode_is ("auto"))
    {
      if ((bb(2)/bb(3)) > (xM/yM))
        // When the image is scaled to the size of the bounding box,
        // the height collides with the bounding box first. Therefore,
        // the camera view angle is defined by the image height yM.
        af = 1.0 / yM;
      else
        // The image width collides with the bounding box.
        af = 1.0 / xM;

      // The view angle "v_angle", also called field of view "FOV",
      // is formed by the hypotenuse and the adjacent side, which is given by
      // the distance between the view point and the camera target "norm (F)".
      // The ratio of the opposite side, given by "af", to the adjacent side in
      // a right-angled triangle is the tangent of the view angle.
      v_angle = 2 * (180.0 / M_PI) * atan (1 / (2 * af * norm (F)));
      cameraviewangle = v_angle;
    }
  else
    v_angle = get_cameraviewangle ();

  // x_projection = diag([1, 1, 1, 1])
  Matrix x_projection = xform_matrix ();
  // Calculate backwards from the angle to the ratio. This step
  // is necessary because "v_angle" can be set manually.
  double pf = 1 / (2 * tan ((v_angle / 2) * M_PI / 180.0) * norm (F));
  // Normalize to one. Resulting coordinates are "normalized device coordinates".
  scale (x_projection, pf, pf, 1);
</syntaxhighlight>}}

"x_viewport" is a 2D transformation used to place the "normalized" plot box in the center of the bounding box:

{{Code|Section of axes::properties::update_camera ()"|<syntaxhighlight lang="C" style="font-size:13px">
  double pix = 1;
  if (autocam)
    {
      if ((bb(2)/bb(3)) > (xM/yM))
        pix = bb(3);
      else
        pix = bb(2);
    }
  else
    pix = (bb(2) < bb(3) ? bb(2) : bb(3));

  // x_viewport = diag([1, 1, 1, 1])
  Matrix x_viewport = xform_matrix ();
  // Move to the center of the bounding box inside the figure.
  translate (x_viewport, bb(0)+bb(2)/2, bb(1)+bb(3)/2, 0);
  // Scale either to width or height, to fit correctly into the bounding box
  scale (x_viewport, pix, -pix, 1);

  x_gl_mat2 = x_viewport * x_projection;
</syntaxhighlight>}}

Note: The matrix just scales x, y so that the image fits tightly into the bounding box. The z-coordinate is not modified.