CSCE 441: Computer Graphics:Hierarchical Models

Jinxiang Chai

Image space

2

Normalized project

space

View spaceWorld spaceObject space

Summary: 3D Geometry Pipeline

Complex Models

Outline

Hierarchical Models

Reading: HB 9-8&9-9, HB chapter 11, OpenGL Programming Guide, chapter 3, and course slides

• Most graphics API supports a few primitives: - sphere - cube - cylinders

• These symbols are instanced using instance/model transformation:

Symbols and Instances

• Most graphics API supports a few primitives: - sphere - cube - cylinders

• These symbols are instanced using instance transformation:

Symbols and Instances

What’s the matrix for the instance transformation above?

• Most graphics API supports a few primitives: - sphere - cube - cylinders

• These symbols are instanced using instance transformation:

Symbols and Instances

),(*)(*),( yxyx ssSRttTM

• In opengl, instance/model transformation is created by modifying the Model-view matrix:

Sample Instance Trans.

glMatrixMode(GL_MODELVIEW);glLoadIdentity(…);// set current matrix to the identityglTranslate(…); // translate glRotate(…); //rotateglScale(…);//scalehouse();

),(*)(*),(* yxyx ssSRttTI

• In opengl, instance transformation is created by modifying the Model-view matrix:

Sample Instance Trans.

Does the transform seem to be backward?

glMatrixMode(GL_MODELVIEW);glLoadIdentity(…);glTranslate(…);glRotate(…);glScale(…);house();

• Opengl postmultiplies transformation matrices as they are called

• Each subsequent transformation call concatenates the designated transformation matrix on the right of the composite matrix

• We must invoke the transformation in the opposite order from which they are applied.

Composite Transformation: Opengl Implementation

glMatrixMode(GL_MODELVIEW);glLoadIdentity(…);M4;M3;M2;M1;…

xMMMMIx *****' 1234

Lamp

• What’s the current coordinate A ?

1

2

3

3c

2c

1c

A

03221100 )()0,()()0,()(),0()(),( pRlTRlTRlTRyxTp

),,( 0yx0c

11

Lamp

• Suppose we know how to draw each bone in local reference frames

A

Lamp

• How can we draw the character under a particular pose

A

Lamp

• What’s the current coordinate A ?

1

2

3

3c

2c

1c

A

03221100 )()0,()()0,()(),0()(),( pRlTRlTRlTRyxTp

),,( 0yx0c

14

Lamp

• What’s the current coordinate A ?

1

2

3

3c

2c

1c

A

03221100 )()0,()()0,()(),0()(),( pRlTRlTRlTRyxTp

),,( 0yx0c

15

Lamp

• What’s the current coordinate A ?

1

2

3

3c

2c

1c

A

03221100 )()0,()()0,()(),0()(),( pRlTRlTRlTRyxTp

),,( 0yx0c

16

Lamp

• What’s the current coordinate A ?

1

2

3

3c

2c

1c

A

03221100 )()0,()()0,()(),0()(),( pRlTRlTRlTRyxTp

),,( 0yx0c

17

Lamp Implementation• The lamp can be displayed by computing a global matrix

and computing it at each step

)(),( 0RyxT

)(),0()(),( 100 RlTRyxT

)0,()()0,()(),0()(),( 221100 lTRlTRlTRyxT

lamp()

{

M_model =

base();

M_model =

upper_arm();

M_model =

middel_arm();

M_model =

lower_arm();}

)()0,()()0,()(),0()(),( 3221100 RlTRlTRlTRyxT

Matrix M_model;

Main()

{ …

M_model=Identity()

lamp();

}

Lamp Implementation• The lamp can be displayed by computing a global matrix

and computing it at each step

)(),( 0RyxT

)(),0()(),( 100 RlTRyxT

)()0,()(),0()(),( 21100 RlTRlTRyxT

lamp()

{

M_model =

base();

M_model =

upper_arm();

M_model =

middel_arm();

M_model =

lower_arm();}

)()0,()()0,()(),0()(),( 3221100 RlTRlTRlTRyxT

Can we make it more efficiently?

Matrix M_model;

Main()

{ …

M_model=Identity()

lamp();

}

Better Implementation• Instead of recalculating the global matrix each time, we

can just update it in place

)(),( 0RyxT

)(),0( 10 RlT

)()0,( 21 RlT

Matrix M_model;

Main()

{ …

M_model=Identity()

lamp();

}

lamp()

{

M_model *=

base();

M_model *=

upper_arm();

M_model *=

middel_arm();

M_model *=

lower_arm();}

)()0,( 32 RlT

Opengl Implementation• Opengl maintains a global state matrix called model-view

matrix

2D_lamp(x, y, θ0, θ1, θ2, θ3){ glTranslatef(x,y,0) glRotatef(θ0,0,0,1,); base(); glTranslatef(0,l0,0) glRotatef(θ1,0,0,1,); upper_arm(); glTranslatef(0,l1,0) glRotatef(θ2,0,0,1,); middel_arm(); glTranslatef(0,l2,0) glRotatef(θ3,0,0,1,); lower_arm();}

Main(){ … glMatrixMode(GL_MODELVIEW); glLoadIdentity();

2D_lamp(a,b,c,d,e,f) …}

//set current matrixto identity

Hierarchical Modeling• Consider a model of a car – how many symbols? – how many instances?

Hierarchical Modeling• Consider a model of a car – 2 symbols : – 5 instances :

Hierarchical Modeling• Consider a model of a car – 2 symbols : chassis + wheel – 5 instances : 1 chassis + 4 wheel

World system

front-left wheel system

Chasis system

Hierarchical Modeling• Consider a model of a car – 2 symbols : chassis + wheel – 5 instances : 1 chassis + 4 wheel

• We can represent our car as a tree to show the relationship between the parts

- tree nodes store subparts - transition edges represent relationship

between the parts

Hierarchical Modeling• Consider a model of a car – 2 symbols : chassis + wheel – 5 instances : 1 chassis + 4 wheel

• We can represent our car as a tree to show the relationship between the parts

• However, since all 4 wheels are instances of the same model, we’d like to only have that model appear once

Hierarchical Modeling• Hierarchical model can be composed of instances using

trees or directed acyclic graphs (DAGs) - edges contains geometric transformations - nodes contains geometry of subparts

Hierarchical Modeling

What might we draw the tree for the lamp?

• Hierarchical model can be composed of instances using trees or directed acyclic graphs (DAGs)

- edges contains geometric transformations - nodes contains geometry of subparts

Hierarchical Modeling• Hierarchical model can be composed of instances using

trees or directed acyclic graphs (DAGs) - edges contains geometric transformations - nodes contains geometry

1

23c

2c

1c

A

Hierarchical Modeling• Hierarchical model can be composed of instances using

trees or directed acyclic graphs (DAGs) - edges contains geometric transformations - nodes contains geometry

lower arm

base

Upper arm

world

)(),( 0RyxT

)(),0( 10 RlT

)()0,( 21 RlT

)()0,( 32 RlTmiddle arm1

23c

2c

1c

A

A More Complex Example: Human Figure

torso

A More Complex Example: Human Figure

torso

A More Complex Example: Human Figure

What’s the most efficient way to draw this figure?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

A More Complex Example: Human Figure

What’s the most sensible way to traverse this tree?

torso

Opengl Implementation: Human Figure

Mh

Mlua

Mlla

torso

Opengl Implementation: Human Figure

Mh

Mlua

Mlla

torso

Is this correct?

Opengl Implementation: Human Figure

Mh

Mh*Mlua should be Mlua

Mh*Mlua*Mlla should be Mlua*Mlla

Itorso

Is this correct? No

Matrix Stack

• glMatrixMode(GL_MODELVIEW) - Initially, each of the stacks contains one matrix, an

identity matrix. - The top matrix in the stack is “current matrix” - The depth is at least 32

Mc

…I

11...MMMM kKc

Push and Pop the Current Matrix

• glPushMatrix() copy the current matrix at the top of the stack

Mc

… Mc

…

Mc

II

Push and Pop the Current Matrix

• glPopMatrix() destroies the matrix at the top of stack

Mtop-1

…

Mtop Mtop-1

…

II

Opengl Implementation: Human Figure

Opengl Implementation: Human Figure

I

Mh

Mlua

Mlua Mlla

Articulated Models