Using Geometric Transformations

Suppose we wish to setup the following transformation:

  M = trans(2,1,0) rot(z,-90) scale(2,2,2)

  glMatrixMode( GL_MODELVIEW );
  glLoadIdentity();
  glTranslatef(2.0, 1.0, 0.0);
  glRotatef( -90, 0.0, 0.0, 1.0); 
  glScalef(2.0, 2.0, 2.0);

  glMatrixMode( GL_MODELVIEW );
  M[0][0] = 2; 
  M[0][[1]= -1;
  ...
  glLoadMatrixf( M );

  glBegin(GL_POLYGON);
    glVertex3fv( vertex_list[i] );
    glVertex3f( x,y,z );
    glVertex2f( x,y );
  glEnd();

Transformation Hierarchies

Many transformations are best specified in a relative fashion.
Consider a car:

  road              with respect to the city
  car               with respect to the road
  front left wheel  with respect to the car
  front rt   wheel  with respect to the car
  rear  left wheel  with respect to the car
  rear  rt wheel    with respect to the car
  hub caps          with respect to wheels

These form a hierarchy of transformations, often called the scene graph. Going down the scene graph is simple -- performing a transformation relative to the current coordinate system simply requires postmultiplication of the current transformation matrix. However, we need a mechanism to return to previous coordinate systems. This is done in OpenGL with the help of a matrix stack and the commands glPushMatrix() and glPopMatrix().