CPSC 441: Computer Graphics Hierarchical Models

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CPSC 441 Computer GraphicsHierarchical Models

• Jinxiang Chai

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Summary 3D Geometry Pipeline
View space
World space
Object space
Normalized project space
Image space
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3
Complex Models
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Outline

• Hierarchical Models
• Animation with motion capture data
• - skeletal model (.asf)
• - motion data (.amc)
• Reading HB chapter 14, OpenGL Programming
  Guide, chapter 3, and course slides

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Symbols and Instances

• Most graphics API supports a few primitives
• - sphere
• - cube
• - cylinders
• These symbols are instanced using instance
  transformation

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Symbols and Instances

• Most graphics API supports a few primitives
• - sphere
• - cube
• - cylinders
• These instances are instanced using instance
  transformation

Whats the matrix for the instance transformation
above?
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Symbols and Instances

• Most graphics API supports a few primitives
• - sphere
• - cube
• - cylinders
• These instances are instanced using instance
  transformation

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Sample Instance Trans.

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

glMatrixMode(GL_MODELVIEW) glLoadIdentity() glT
ranslate() glRotate() glScale() house()
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Sample Instance Trans.

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

glMatrixMode(GL_MODELVIEW) glLoadIdentity() glT
ranslate() glRotate() glScale() house()
Does the transform seem to be backward?
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Composite transformation opengl

• 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.

glMatrixMode(GL_MODELVIEW) glLoadIdentity() M4
M3 M2 M1
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3D Example A Robot Arm

• Consider a robot arm with 3 degrees of freedom
• - Base rotates about vertical axis by ?
• - Lower arm rotates about z axis by F
• - Upper arm rotates about z axis by ?

lower arm
Upper arm
h2
h3
h1
base
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3D Example A Robot Arm

• Consider a robot arm with 3 degrees of freedom
• - Base rotates about vertical axis by ?
• - Lower arm rotates about z axis by F
• - Upper arm rotates about z axis by ?

lower arm
Upper arm
h2
h3
h1
base
What are matrices we use to transform base, upper
arm and lower arm respectively?
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3D Example A Robot Arm

• Consider a robot arm with 3 degrees of freedom
• - Base rotates about vertical axis by ?
• - Lower arm rotates about z axis by F
• - Upper arm rotates about z axis by ?

lower arm
Upper arm
base
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3D Example A Robot Arm

• Consider a robot arm with 3 degrees of freedom
• - Base rotates about vertical axis by ?
• - Lower arm rotates about z axis by F
• - Upper arm rotates about z axis by ?

base
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3D Example A Robot Arm

• Consider a robot arm with 3 degrees of freedom
• - Base rotates about vertical axis by ?
• - Lower arm rotates about z axis by F
• - Upper arm rotates about z axis by ?

Upper arm
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3D Example A Robot Arm

• Consider a robot arm with 3 degrees of freedom
• - Base rotates about vertical axis by ?
• - Lower arm rotates about z axis by F
• - Upper arm rotates about z axis by ?

lower arm
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Robot Arm Implementation

• The robot arm can be displayed by computing a
  global matrix and computing it at each step

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Robot Arm Implementation

• The robot arm can be displayed by computing a
  global matrix and computing it at each step

Can we make it more efficient?
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Better Implementation

• Instead recalculating the global matrix each
  time, we can just update it in place

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Opengl Implementation

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

//set current matrix to identity
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Another Example

• A 2D lamp with 6 degrees of freedom

lower arm
middle arm
Upper arm
base
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Another Example

• A 2D lamp with 6 degrees of freedom

base
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Another Example

• A 2D lamp with 6 degrees of freedom

Upper arm
base
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Another Example

• A 2D lamp with 6 degrees of freedom

middle arm
Upper arm
base
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Another Example

• A 2D lamp with 6 degrees of freedom

lower arm
middle arm
Upper arm
base
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Another Example

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

2D_lamp(x, y, ?0, ?1, ?2, ?3)
glTranslatef(x,y,0) glrotatef(?0,0,0,1,)
base() glTranslatef(0,l0,0)
glrotatef(?1,0,0,1,)
upper_arm() glTranslatef(0,l1,0)
glrotatef(?2,0,0,1,)
middel_arm() glTranslatef(0,l2,0)
glrotatef(?3,0,0,1,)
lower_arm()
Main() glMatrixMode(GL_MODELVIEW)
glLoadIdentity() 2D_lamp(a,b,c,d,e,f)

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Hierarchical Modeling

• Consider a model of a car
• 2 symbols
• 5 instances

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Hierarchical Modeling

• Consider a model of a car
• 2 symbols chassis wheel
• 5 instances 1 chassis 4 wheel

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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

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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, wed like to only have that model
  appear once

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Hierarchical Modeling

• Hierarchical model can be composed of instances
  using trees or directed acyclic graphs (DAGs)
• - edges contains geometric transformations
• - nodes contains geometry

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Hierarchical Modeling

• Hierarchical model can be composed of instances
  using trees or directed acyclic graphs (DAGs)
• - edges contains geometric transformations
• - nodes contains geometry

What might we draw the tree for the robot arm?
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Hierarchical Modeling

• Hierarchical model can be composed of instances
  using trees or directed acyclic graphs (DAGs)
• - edges contains geometric transformations
• - nodes contains geometry

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Hierarchical Modeling

• Hierarchical model can be composed of instances
  using trees or directed acyclic graphs (DAGs)
• - edges contains geometric transformations
• - nodes contains geometry

world
base
Upper arm
lower arm
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A More Complex Example Human Figure
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A More Complex Example Human Figure
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A More Complex Example Human Figure
Whats the most efficient way to draw this figure?
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A More Complex Example Human Figure
Whats the most sensible way to traverse this
tree?
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A More Complex Example Human Figure
Whats the most sensible way to traverse this
tree?
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A More Complex Example Human Figure
Whats the most sensible way to traverse this
tree?
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A More Complex Example Human Figure
Whats the most sensible way to traverse this
tree?
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A More Complex Example Human Figure
Whats the most sensible way to traverse this
tree?
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A More Complex Example Human Figure
Whats the most sensible way to traverse this
tree?
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A More Complex Example Human Figure
Whats the most sensible way to traverse this
tree?
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A More Complex Example Human Figure
Whats the most sensible way to traverse this
tree?
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A More Complex Example Human Figure
Whats the most sensible way to traverse this
tree?
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A More Complex Example Human Figure
Whats the most sensible way to traverse this
tree?
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Opengl Implementation Human Figure
Mh
Mlua
Mlla
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Opengl Implementation Human Figure
I
Mh
MhMlua
MhMluaMlla
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Matrix Stack

• glMatrixMode(GL_MODELVIEW)
• - Initially, each of the stacks contains one
  matrix, an identity matrix.
• - The depth is at least 32

M

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Push and Pop the Current Matrix

• glPushMatrix
• glPopMatrix

M
M

M

M2
M1
M2


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Opengl Implementation Human Figure
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Opengl Implementation Human Figure
I
Mh
Mlua
Mlua Mlla
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Outline

• Hierarchical Models
• Animation with motion capture data
• - skeletal model (.asf)
• - motion data (.amc)

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3D Human Characters

• Articulated Human models
• - rigid parts
• - connected by joints
• They can be animated by specifying the joint
  angles (or other display parameters) as functions
  of time.

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Human motion representation
A sequence of poses q1,q2,qT Each pose is
represented as a high-dimensional vector qt
Rn
Motion trajectories
Pose qt
Motion q1,qT
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Human-body Animation

Movies!
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Motion capture data files

• Each sequence of human motion data contains two
  files
• Skeleton file (.asf) Specify the skeleton model
  of character
• Motion data file (.amc) Specify the joint angle
  values over the frame/time
• Both files are generated by Vicon software

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Human skeletal file
Described in a default pose
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Human skeletal model
This is still a tree!
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Human skeletal file (.asf)

• individual bone information
• - length of the bone
• - direction of the bone
• - local coordinate frame
• - number of Dofs
• - joint limits
• bone hierarchy/connections

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Individual bone information

• begin id bone_id                  / Unique
  id for each bone /name bone_name        /
  Unique name for each bone /direction dX dY
  dZ    / Vector describing direction of the
  bone in world / coor. system length
  7.01722           / Length of the bone/
  axis 0 0 20 XYZ         / Rotation of local
  coordinate system for                            
          this bone relative to the world
  coordinate                                   
  system. In .AMC file the rotation angles
                                      for this
  bone for each time frame will be
                                     defined
  relative to this local coordinate
                                      system/
  dof rx ry rz                / Degrees of
  freedom for this bone. limits (-160.0 20.0)
  / joint limits/            (-70.0 70.0)
              (-60.0 70.0) end

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Individual bone information
begin id 2                 name
lfemur        direction 0.34 -0.93 0   
length 7.01722           axis 0 0 20
XYZ         dof rx ry rz               
limits (-160.0 20.0)             (-70.0 70.0)
            (-60.0 70.0) end
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Individual bone information
begin id 2                 name
lfemur        direction 0.34 -0.93 0   
length 7.01722           axis 0 0 20
XYZ         dof rx ry rz               
limits (-160.0 20.0)             (-70.0 70.0)
            (-60.0 70.0) end
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Individual bone information
begin id 2                 name
lfemur        direction 0.34 -0.93 0   
length 7.01722           axis 0 0 20
XYZ         dof rx ry rz               
limits (-160.0 20.0)             (-70.0 70.0)
            (-60.0 70.0) end
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Individual bone information
begin id 2                 name
lfemur        direction 0.34 -0.93 0   
length 7.01722           axis 0 0 20
XYZ         dof rx ry rz               
limits (-160.0 20.0)             (-70.0 70.0)
            (-60.0 70.0) end
yk
xk
zk
Euler angle representation
RkRz(?)Ry(ß)Rx(a)
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Individual bone information
begin id 2                 name
lfemur        direction 0.34 -0.93 0   
length 7.01722           axis 0 0 20
XYZ         dof rx ry rz               
limits (-160.0 20.0)             (-70.0 70.0)
            (-60.0 70.0) end
yk
xk
zk
- The number of dof for this joint - The minimal
and maximum joint angle for each dof
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Individual bone information
begin id 2                 name
lfemur        direction 0.34 -0.93 0   
length 7.01722           axis 0 0 20
XYZ         dof rx ry rz               
limits (-160.0 20.0)             (-70.0 70.0)
            (-60.0 70.0) end
yk
1-dof joint
2-dof joint
3-dof joint
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Individual bone information
begin id 2                 name
lfemur        direction 0.34 -0.93 0   
length 7.01722           axis 0 0 20
XYZ         dof rx ry rz               
limits (-160.0 20.0)             (-70.0 70.0)
            (-60.0 70.0) end
yk
yk1
Xk1
zk1
begin id 3                 name
ltibia        direction 0.34 -0.93 0   
length 7.2138           axis 0 0 20
XYZ         dof rx            limits
(-10.0 170.0) end
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Root representation

• root
• order TX TY TZ RX RY RZ
• axis XYZ
• position 0 0 0
• orientation 0 0 0

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Root representation

• root
• order TX TY TZ RX RY RZ
• axis XYZ
• position 0 0 0
• orientation 0 0 0

How to compute the coordinate of a joint in the
world coordinate frame?
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Root representation

• root
• order TX TY TZ RX RY RZ
• axis XYZ
• position 0 0 0
• orientation 0 0 0

How to compute the coordinate of a joint in the
world coordinate frame?
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Hierarchy/Bone Connections
hierarchy begin root lhipjoint rhipjoint
lowerback lhipjoint lfemur lfemur ltibia
ltibia lfoot lfoot ltoes rhipjoint
rfemur rfemur rtibia rtibia rfoot
rfoot rtoes lowerback upperback upperback
thorax thorax lowerneck lclavicle rclavicle
end
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Hierarchy/Bone Connections
hierarchy begin root lhipjoint rhipjoint
lowerback lhipjoint lfemur lfemur ltibia
ltibia lfoot lfoot ltoes rhipjoint
rfemur rfemur rtibia rtibia rfoot
rfoot rtoes lowerback upperback upperback
thorax thorax lowerneck lclavicle rclavicle
end
lowerback
root
rhipjoint
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Hierarchy/Bone Connections
hierarchy begin root lhipjoint rhipjoint
lowerback lhipjoint lfemur lfemur ltibia
ltibia lfoot lfoot ltoes rhipjoint
rfemur rfemur rtibia rtibia rfoot
rfoot rtoes lowerback upperback upperback
thorax thorax lowerneck lclavicle rclavicle
end
lowerback
root
rhipjoint
lhipjoint
lfemur
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Hierarchy/Bone Connections
hierarchy begin root lhipjoint rhipjoint
lowerback lhipjoint lfemur lfemur ltibia
ltibia lfoot lfoot ltoes rhipjoint
rfemur rfemur rtibia rtibia rfoot
rfoot rtoes lowerback upperback upperback
thorax thorax lowerneck lclavicle rclavicle
end
lowerback
root
rhipjoint
lhipjoint
lfemur
ltibia
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Hierarchy/Bone Connections
hierarchy begin root lhipjoint rhipjoint
lowerback lhipjoint lfemur lfemur ltibia
ltibia lfoot lfoot ltoes rhipjoint
rfemur rfemur rtibia rtibia rfoot
rfoot rtoes lowerback upperback upperback
thorax thorax lowerneck lclavicle rclavicle
end
lowerback
root
rhipjoint
lhipjoint
lfemur
ltibia
lfoot
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Hierarchy/Bone Connections
hierarchy begin root lhipjoint rhipjoint
lowerback lhipjoint lfemur lfemur ltibia
ltibia lfoot lfoot ltoes rhipjoint
rfemur rfemur rtibia rtibia rfoot
rfoot rtoes lowerback upperback upperback
thorax thorax lowerneck lclavicle rclavicle
end
lowerback
root
rhipjoint
lhipjoint
lfemur
ltibia
lfoot
ltoe
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What can we do with .asf file?

• We can visualize the default pose
• We can compute various transforms in the default
  pose
• - between world coordinate frame and local
  coordinate
• - between parent coordinate frame and child
  coordinate frame

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From local coordinate to world coordinate
yk
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From local coordinate to world coordinate
yk
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From child to parent node

• How to Compute the transformation Tkk-1 from a
  child local coordinate frame to its parent local
  coordinate frame

Tkk-1
x
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Bone transform
parent
Tkk-1?
world
child
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Bone transform
parent
Tkk-1?
world
child
Given Rk,Tk, Rk-1,Tk-1, - How can we compute
xk-1 from xk?
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Bone transform
parent
Tkk-1?
world
child
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Bone transform
parent
Tkk-1?
world
child
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Motion data file (.amc)

• i
  // frame
  number
• root 2.36756 16.4521 12.3335 -165.118 31.188
  -179.889 // root position and orientation
• lowerback -17.2981 -0.243065 -1.41128
  // joint angles for lowerback joint
• upperback 0.421503 -0.161394 2.20925
  // joint angles for thorax joint
• thorax 10.2185 -0.176777 3.1832
• lowerneck -15.0172 -5.84786 -7.55529
• upperneck 30.0554 -3.19622 -4.68899
• head 12.6247 -2.35554 -0.876544
• rclavicle 4.77083e-014 -3.02153e-014
• rhumerus -23.3927 30.8588 -91.7324
• rradius 108.098
• rwrist -35.4375
• rhand -5.30059 11.2226
• rfingers 7.12502
• rthumb 20.5046 -17.7147
• lclavicle 4.77083e-014 -3.02153e-014
• lhumerus -35.2156 -19.5059 100.612

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Motion data file (.amc)

• i
  // frame
  number
• root 2.36756 16.4521 12.3335 -165.118 31.188
  -179.889 // root position and orientation
• lowerback -17.2981 -0.243065 -1.41128
  // joint angles for lowerback joint
• upperback 0.421503 -0.161394 2.20925
  // joint angles for thorax joint
• thorax 10.2185 -0.176777 3.1832
• lowerneck -15.0172 -5.84786 -7.55529
• upperneck 30.0554 -3.19622 -4.68899
• head 12.6247 -2.35554 -0.876544
• rclavicle 4.77083e-014 -3.02153e-014
• rhumerus -23.3927 30.8588 -91.7324
• rradius 108.098
• rwrist -35.4375
• rhand -5.30059 11.2226
• rfingers 7.12502
• rthumb 20.5046 -17.7147
• lclavicle 4.77083e-014 -3.02153e-014
• lhumerus -35.2156 -19.5059 100.612

- Rotation described in local coordinate frame -
Euler angle representation x-y-z
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Composite 3D Transformation
From .asf file
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Composite 3D Transformation
From .amc file
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Composite 3D Transformation
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Composite 3D Transformation
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Composite 3D Transformation
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Composite 3D Transformation
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Human Figures

• Traverse the whole tree to draw every bones

This is still a tree!