Chapter 3:

Modeling Transformation

Graphics Programming, 8th Sep.

Graphics and Media Lab.

Seoul National University 2011 Fall

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• Every step in the graphics pipeline is related to the transformation.

OpenGL Steps

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• Euclidean (Rigid) motion– Preserves distance & angle.

• Similarity motion– Preserves angle.

• Linear motion– Preserves linearity.

Related Transformations (1/2)

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• Affine motion– Preserves parallel line.

• Projective motion– Preserves line.

How to represent ?

Related Transformations (2/2)

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• Euclidean transformation(SE3)

– Preserves distance & angle.(Translation, Rotation)

• Similarity transformation– Preserves angles.(Rotation, Isotropic scaling)

• Linear transformation(GL3)

– Preserves linearity.(Rotation,Scaling,Reflection,Shear)

• Affine transformation– Preserves parallel lines.

• Projective transformation– Preserves lines.

4x4 Matrix !

How about 3D points ?

Transformation Classes

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• We augment a point in R3 for the transformation matrices.– Modelview matrix has the form

– Projection matrix has the form (The next class!)

– Point [x y z 1]T

• The result of transformation needs to be divided by w to give the 3D position. (homogeneous)

– Vector [x y z 0]T

• w=0 represents a point at “ infinity”.

(Only direction differentiates.)

Homogeneous Coordinates

1000

3222120

2121110

1020100

taaa

taaa

taaa

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• OpenGL helps us to change the two most important transformation matrices:– Modelview Matrix

• The relative transformation between object and camera

– Projection Matrix

• Clipping volume (viewing frustum)

• Projection to screen

• Vertices(primitives) are transformed by P*M.

Modelview & Projection matrix

P M* *

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• Draw a transformed box in 3d.

Code Example from Ch.2

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• Resize() function was added.

• A few experiments with modeling transform

Code Example 1

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Result

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• glutMainLoop() instructs the program to enter the event processing loop.

• glutReshapeFunc() registers the resizing function as the callback function.

• glClear(…) clears video buffers.

• glViewport(x,y,w,h) sets the screen space.

• glMatrixMode(GL_PROJECTION) tells that we modify the projection matrix.

• glMatrixMode(GL_MODELVIEW) tells that we modify the modelviewmatrix.

• glLoadIdentity() replaces the current matrix with the identity matrix.

• glOrtho(l,r,b,t,zn,zf) sets the clipping space orthographically. This changes the projection matrix.

• glTranslatef(),glRotatef(), and glScalef() change the modelview matrix.

Code Example 1 - Explanation

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glLoadIdentity(); // C = I

glMultMatrixf(N); // C = N

glMultMatrixf(M); // C = NM

glMultMatrixf(L); // C = NML

glBegin(GL_POINTS);

glVertex3f(v); // NMLv

glEnd();

The Order of Transformations

Application

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• 45 deg rotation around z-axis then 10 unit translation along +x, and vice versa.

The Order is Important

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• Code for rot then trans wrt fixed coords.

glMatrixMode(GL_MODELVIEW);

glLoadIdentity();

glMultMatrixf(T);

glMultMatrixf(R);

draw_the_object();

Thinking in Terms of Fixed Coords

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• Code for trans then rot wrt local coords.

glMatrixMode(GL_MODELVIEW);

glLoadIdentity();

glMultMatrixf(T);

glMultMatrixf(R);

draw_the_object();

Thinking in Terms of Local Coords

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• Code for rot then trans wrt local coords.

glMatrixMode(GL_MODELVIEW);

glLoadIdentity();

glMultMatrixf(R);

glMultMatrixf(T);

draw_the_object();

Thinking in Terms of Local Coords

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• We can manage the hierarchy by glPushMatrix(), glPopMatrix().

Push & Pop

P M*

M1

Vertices

of shoulder

Vertices

of body

Vertices

of arm

M2

* * *

Push

Push

Pop

Pop

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• Robot arm

Code Example 2

unit cube: [-0.5,0.5]^3

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Result

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• glutKeyboardFunc() registers the key function as the callback function:– If „s‟ is pressed, rotate shoulder– If „S‟ is pressed, rotate shoulder reversely– If „e‟ is pressed, rotate elbow– If „E‟ is pressed, rotate elbow reversely

• glPushMatrix() saves the current matrix.

• glPopMatrix() loads the saved matrix.

• gluPerpective(fov,ratio,zn,zf) sets the view frustum. This changes the projection matrix.

• glTranslatef(0,0,-1);glRotatef(-angle,0,0,1);glTranslatef(0,0,1);– The rotation about a point

Code Example 2 - Explanation