Scene Graph - Introduction to Computer Graphics - Lecture Slides, Slides of Computer Graphics

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Lecture 14:

Scene Graph

Refresher: OpenGL Matrix

Transformation Pipeline

• Input: list of 3D coordinates (x, y, z)
• GL_MODELVIEW
  • Model transform (earlier)
  • View transform
• GL_PROJECTION
  • Projection transform
  • Clipping
  • Perspective division
• Viewport
  • Viewport transform
• Output: 2D coordinates for each

input 3D coordinates

In Summary!

• The full projection matrix: 𝑀𝑝𝑝𝑆𝑥𝑦𝑧𝑅𝑢𝑣𝑧2𝑥𝑦𝑧𝑇𝑢𝑣𝑤:

1

tan 𝜃𝑤 2 𝑓𝑎𝑟

0 0 0

0

1

tan 𝜃 2 ℎ 𝑓𝑎𝑟

0 0

0 0

1 𝑓𝑎𝑟

0 0 0 0 1

• Important!! Remember to homogenize your points

after you apply this transform

Putting Everything All Together

• We know about camera and object modeling transformations now, let’s

put them together:

• 1. 𝑁𝑝𝑒𝑟𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒 = 𝑀𝑝𝑝𝑀𝑝𝑒𝑟𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒
• 2. 𝐶𝑀𝑇𝑀 = 𝑆𝑅𝑇
  • The CMTM (Composite Modeling Transformation Matrix) is a composite matrix

of all of our object modeling transformations (Scaling, Rotating, Translations,

etc)

• 3. 𝐶𝑇𝑀 = 𝑁𝑝𝑒𝑟𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒 ∗ 𝐶𝑀𝑇𝑀
  • The CTM (Composite Transformation Matrix) is the combination of all our

camera and modeling transformations

• In OpenGL it is referred to as the ModelViewProjection Matrix
• Model: Modeling Transformations
• View: Camera translate/rotate
• Projection: Frustum scaling/unhinging

Applying Composite Matrices

Matrix M = T * (R * S);

for (i=0; i<numTriangles; i++) {

Triangle t = MyObject->triangles[i]; glBegin(GL_TRIANGLES); for (j=0; j<3; j++) { Position p = t->vertex[j]->GetPosition(); Position newP = M * p; glVertex3d (newP.x, newP.y, newP.z); } glEnd();

}

• Yikes! Matrix multiplication in software. SLOW!

Applying Composite Matrices with

OpenGL

Matrix M = T * (R * S);

glLoadMatrixd(M.getValues());

for (i=0; i<numTriangles; i++) {

Triangle t = MyObject->triangles[i];

glBegin(GL_TRIANGLES);

for (j=0; j<3; j++) {

Position p = t->vertex[j]->GetPosition();

glVertex3d (newP.x, newP.y, newP.z);

glEnd();

• Much better… Now we’re using the graphics card to do the

computation for us.

Be careful! Read the

OpenGL spec to see

how you are

arranging your

matrix orientation

and make sure that

it’s consistent with

OpenGL.

You might need to

do a transpose

Drawing Multiple Objects

• Let’s say that you now have 2 objects, myCube

and mySphere.

• Each object needs to be transformed

differently:

– For myCube, the transforms are 𝑀𝑐 = 𝑇𝑐𝑅𝑐𝑆𝑐

– For mySphere, the transforms are: 𝑀𝑠 = 𝑇𝑠𝑅𝑠𝑆𝑠

• Let’s see how we will write this code:

Drawing Multiple Objects

glMatrixMode(GL_PROJECTION);

glLoadMatrixd(myCamera->getProjectionMatrix().getValues());

glMatrixMode(GL_MODELVIEW);

glLoadMatrixd(myCamera->getModelViewMatrix().getValues());

Matrix Mc = Tc * (Rc * Sc);

glMultMatrixd(Mc.getValues());

myCube.draw();

Matrix Ms = Ts * (Rs * Ss);

glMultMatrixd(Ms.getValues());

mySpher.draw();

Drawing Multiple Objects

glMatrixMode(GL_PROJECTION);

glLoadMatrixd(myCamera->getProjectionMatrix().getValues());

glMatrixMode(GL_MODELVIEW);

glLoadMatrixd(myCamera->getModelViewMatrix().getValues());

Matrix Mc = Tc * (Rc * Sc);

glMultMatrixd(Mc.getValues());

myCube.draw();

Matrix Mc_Inv = Mc.Inverse();

glMultMatrixd(Mc_Inv.getValues());

Matrix Ms = Ts * (Rs * Ss);

glMultMatrixd(Ms.getValues());

mySpher.draw();

What about…

Drawing Multiple Objects

glMatrixMode(GL_PROJECTION);

glLoadMatrixd(myCamera->getProjectionMatrix().getValues());

glMatrixMode(GL_MODELVIEW);

glLoadMatrixd(myCamera->getModelViewMatrix().getValues());

glPushMatrix();

Matrix Mc = Tc * (Rc * Sc);

glMultMatrixd(Mc.getValues());

myCube.draw();

glPopMatrix();

glPushMatrix();

Matrix Ms = Ts * (Rs * Ss);

glMultMatrixd(Ms.getValues());

mySpher.draw();

glPopMatrix();

Hierarchical Transform

• 3D objects are often constructed in a way that

have dependency

– This means that an object’s position, orientation,

etc. will depend on its parents position,

orientation, etc.

upper arm

Hierarchical Transform

• While you can compute how the hammer will

move based on lower arm’s rotation, this is

really a pain.

– You’ll need to calculate the rotation of the lower

arm.

– Followed by the upper arm.

– And finally the hammer. upper arm

Basic Approach

• Let’s say that you want

to move the Robotic Arm

to the right (in +x axis)

by 5 units, and the upper

arm rotated by some

amount.

– Using the Scene Graph,

this could mean that we

iteratively apply:

• The same Translation to

all the children of the root

node (base)

• The same Rotation to all

the children of the upper

arm

Relative Transformation

• In this sense, all the transformations are

relative to an object’s parents / ancestors.