Hierarchical transformations

Graphics & Media Lab. >> Êóðñû >> Êóðñ Þ.Ì.Áàÿêîâñêîãî 1999

Using Geometric Transformations

The simplest method of modelling objects is to use primitives such as lines and polygons. In the following, M is the desired transformation matrix which transforms points (2D or 3D) to pixel coordinates.

  for each vertex i

    new_vertex_list[i] = M * vertex_list[i]

  scanconvert( new_vertex_list )

Here is the equivalent in OpenGL:.

  glBegin(GL_POLYGON);

  for each vertex i

    glVertex3fv( vertex_list[i] );

  glEnd();

There are several things to note:

glVertex3fv() :

multiplies vertex by M
feeds points to scan conversion

glBegin() : specifies drawing mode

GL_POINTS
GL_LINE_LOOP
GL_POLYGON
...

matrix M is hidden ( current transformation matrix)

M can be setup as follows:

  glMatrixMode( GL_MODELVIEW );

  glLoadIdentity();

  glTranslatef(2.0, 1.0, 0.0);

  glRotatef( -3.14/2.0, 0.0, 0.0, 1.0); 

  glScalef(2.0, 2.0, 2.0);

  ...

which produces the matrix M = trans(2,1,0) rot(z,-90) scale(2,2,2) ....
Another way of loading M is to use:

  glMatrixMode( GL_MODELVIEW );

  glLoadMatrixf( M );

Transformation Hierarchies

Consider building the following model of a hand with one finger::

This can be constructed using a transformation hierarchy. In the following scene graph, circles represent transformations and squares represent geometry. The pseudocode on the left can be used to draw the scene.

f1: trans(d_hand,0,0) rot(z,th1)
f2: trans(d1,0,0) rot(z,th2)
f3: trans(d2,0,0) rot(z,th3)

M=M*Thand
draw hand
M=M*Tf1
draw f1
M=M*Tf2
draw f2
M=M*Tf3
draw f3

Now consider draw a hand with three identical fingers. We can create a more complex scene graph which uses multiple instances of a finger scene graph, as shown below. Because each of the fingers is defined relative to the hand coordinate system, a way is needed to restore the hand coordinate system before beginning to draw each finger. This is done through the pushMatrix() and popMatrix() function calls.

M=M*Thand
draw hand
pushMatrix()
M=M*Tf1a
draw_finger()
popMatrix()
pushMatrix()
M=M*Tf1b
draw_finger()
popMatrix()
pushMatrix()
M=M*Tf1c
draw_finger()
popMatrix()
draw_finger() {
draw f1
M=M*Tf2
draw f2
M=M*Tf3
draw f3
}

Many graphics systems maintain a stack for the current transformation matrix:

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