Modeling Transformations

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Modeling Transformations

• 2D Transformations
• 3D Transformations
• OpenGL Transformation

2
2D-Transformations

• Basic Transformations
• Homogeneous coordinate system
• Composition of transformations

3
Translation 2D
4
Scaling 2D

• Types of Scaling
• Differential ( sx ! sy )
• Uniform ( sx sy )

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Rotation 2D
6
Rotation 2D
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Mirror Reflection
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Shearing Transformation
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Inverse 2D - Transformations
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Homogeneous Co-ordinates

• Translation, scaling and rotation are expressed
  (non-homogeneously) as
• translation P? P T
• Scale P? S P
• Rotate P? R P
• Composition is difficult to express, since
  translation not expressed as a matrix
  multiplication
• Homogeneous coordinates allow all three to be
  expressed homogeneously, using multiplication by
  3 3 matrices
• W is 1 for affine transformations in graphics

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Homogeneous Co-ordinates

• P2d is a projection of Ph onto the w 1 plane
• So an infinite number of points correspond to
  they constitute the whole line (tx, ty, tw)

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Classification of Transformations

• Rigid-body Transformation
• Preserves parallelism of lines
• Preserves angle and length
• e.g. any sequence of R(?) and T(dx,dy)
• Affine Transformation
• Preserves parallelism of lines
• Doesnt preserve angle and length
• e.g. any sequence of R(?), S(sx,sy) and T(dx,dy)

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Properties of rigid-body transformation
The following Matrix is Orthogonal if the upper
left 2X2 matrix has the following properties

• A) Each row are unit vector.
• sqrt(r11 r11 r12 r12) 1
• B) Each column are unit vector.
• sqrt(c11 c11 c12 c12) 1
• 2. A) Rows will be perpendicular to each other
• (r11 , r12 ) . ( r21 , r22) 0
• B) Columns will be perpendicular to each other
• (c11 , c12 ) . (c21 ,c22) 0
• e.g. Rotation matrix is orthogonal

• Orthogonal Transformation ? Rigid-Body
  Transformation
• For any orthogonal matrix B ? B-1 BT

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Commutativity of Transformation Matrices

• In general matrix multiplication is not
  commutative
• For the following special cases commutativity
  holds i.e. M1.M2 M2.M1

M1 M2
Translate Translate
Scale Scale
Rotate Rotate
Uniform Scale Rotate

• Some non-commutative Compositions
• Non-uniform scale, Rotate
• Translate, Scale
• Rotate, Translate

Original Transitional Final
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Associativity of Matirx Multiplication
Create new affine transformations by multiplying
sequences of the above basic transformations.
q CBAp q ( (CB) A) p (C (B A))p C (B
(Ap) ) etc. matrix multiplication is
associative.
To transform just a point, better to do q
C(B(Ap))
But to transform many points, best to do
M CBA
then do q Mp for any
point p to be rendered.
For geometric pipeline transformation, define M
and set it up with the model-view matrix and
apply it to any vertex subsequently defined to
its setting.
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Rotation of ? about P(h,k) R?,P
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Scaling w.r.t. P(h,k) Ssx,sy,p
T(-h ,-k)
S(sx,sy)
T(h ,k)
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Reflection about line L, ML
Step 1 Translate (0,b) to origin
Step 2 Rotate -? degrees
Step 3 Mirror reflect about X-axis
Step 4 Rotate ? degrees
Step 5 Translate origin to (0,b)
T(0 ,-b)
ML
R(-?)
T(0 ,b)
M x
R(?)
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Problems to be solved

• Schaums outline series
• Problems
• 4.1
• 4.2
• 4.3, 4.4, 4.5 gt R?,P
• 4.6, 4.7, 4.8 gt Ssx,sy,,P
• 4.9, 4.10, 4.11, 4.21 gt ML
• 4.12 gt Shearing
• Pg-281(1.24), Pg-320(5.19)
• gt Circular view-port

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3D Transformations

• Basics of 3D geometry
• Basic 3D Transformations
• Composite Transformations

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Orientation

• Thumb points to ve Z-axis
• Fingers show ve rotation from X to Y axis

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Vectors in 3D

• Have length and direction
• V xv , yv , zv
• Length is given by the Euclidean Norm
• V ?( xv2 yv2 zv2 )
• Dot Product
• V U xv, yv, zvxu, yu, zu
• xvxu yvyu zvzu
• V U cos ß
• Cross Product
• V ? U
• yvzu - zv yu , -xvzu zvxu , xvyu yvxu
  
• ? V U sin ß
• V ? U - ( U x V)

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3D Equation of Curve Line

• Parametric equations of Curve Line
• Curve
• Line

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3D Equation of Surface Plane

• Parametric equations of Surface Plane
• Surface
• Plane with Normal, N

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3D Plane

• Ways of defining a plane
• 1. 3 points P0, P1, P2 on the plane
• 2. Plane Normal N P0 on plane
• 3. Plane Normal N a vector V on the plane
• Plane Passing through P0, P1, P2

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Affine Transformation

• Transformation is a function that takes a point
  (or vector) and maps that point (or vector) into
  another point (or vector).
• A coordinate transformation of the form
• x axx x axy y axz z bx ,
• y ayx x ayy y ayz z by ,
• z azx x azy y azz z bz ,
• is called a 3D affine transformation.

• The 4th row for affine transformation is always
  0 0 0 1.
• Properties of affine transformation
• translation, scaling, shearing, rotation (or any
  combination of them) are examples affine
  transformations.
• Lines and planes are preserved.
• parallelism of lines and planes are also
  preserved, but not angles and length.

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Translation 3D
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Scaling 3D
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Rotation 3D

• For 3D-Rotation 2 parameters are needed
• Angle of rotation
• Axis of rotation

Rotation about z-axis
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Rotation about Y-axis X-axis
About y-axis
About x-axis
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Shear along Z-axis
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Object Transformation

• Line Can be transformed by transforming the end
  points
• Plane(described by 3-points) Can be transformed
  by transforming the 3-points
• Plane(described by a point and Normal) Point is
  transformed as usual. Special treatment is needed
  for transforming Normal

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Composite Transformations 3D

• Some of the composite transformations to be
  studied are
• AV,N aligning a vector V with a vector N
• R?,L rotation about an axis L( V, P )
• Ssx,sy,P scaling w.r.t. point P

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AV aligning vector V with k
Av
R?,i
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AV aligning vector V with k
Av
R?,i
R-?,j
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AV aligning vector V with k

• AV-1 AVT
• AV,N AN-1 AV

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R?,L rotation about an axis L

• Let the axis L be represented by vector V and
  passing through point P

1. Translate P to the origin

2. Align V with vector k
3. Rotate ?? about k
4. Reverse step 2
5. Reverse step 1
R?,L
T-P
AV
R?,k
AV-1
T-P-1
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MN,P Mirror reflection

• Let the plane be represented by plane normal N
  and a point P in that plane

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MN,P Mirror reflection

• Let the plane be represented by plane normal N
  and a point P in that plane

1. Translate P to the origin

MN,P
T-P
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MN,P Mirror reflection

• Let the plane be represented by plane normal N
  and a point P in that plane

1. Translate P to the origin

N
2. Align N with vector k
P
MN,P
T-P
AN
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MN,P Mirror reflection

• Let the plane be represented by plane normal N
  and a point P in that plane

1. Translate P to the origin

2. Align N with vector k
3. Reflect w.r.t xy-plane
MN,P
T-P
AN
S1,1,-1
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MN,P Mirror reflection

• Let the plane be represented by plane normal N
  and a point P in that plane

1. Translate P to the origin

2. Align N with vector k
3. Reflect w.r.t xy-plane
4. Reverse step 2
MN,P
T-P
AN
S1,1,-1
AN-1
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MN,P Mirror reflection

• Let the plane be represented by plane normal N
  and a point P in that plane

1. Translate P to the origin

2. Align N with vector k
3. Reflect w.r.t xy-plane
4. Reverse step 2
5. Reverse step 1
MN,P
T-P
AN
S1,1,-1
AN-1
T-P-1
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Further Composition
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Finding R
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Finding Rz
R
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Finding Rx
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Finding Ry
Rz
Ry
Rx
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Problems to be solved

• Schaums outline series
• Problems
• 6.1
• 6.2, 6.5, 6.9, 6.10, 6.11, 6.12 ? Av
• 6.3, 6.4 ? R?,L
• 6.6, 6.7, 6.8 ? MN,P

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Transformations in OpenGL

• OpenGL transformation commands
• Transformation Order
• Hierarchical Modeling

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Transformations in OpenGL

• OpenGL uses 3 stacks to maintain transformation
  matrices
• Model View transformation matrix stack
• Projection matrix stack
• Texture matrix stack
• You can load, push and pop the stack
• The top most matrix from each stack is applied to
  all graphics primitive until it is changed

M
N
Graphics Primitives (P)
Output NMP
Projection Matrix Stack
Model-View Matrix Stack
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General Transformation Commands

• Specify current matrix (stack)
• void glMatrixMode(GLenum mode)
• Mode GL_MODELVIEW, GL_PROJECTION, GL_TEXTURE
• Initialize current matrix.
• void glLoadIdentity(void)
• Sets the current matrix to 4X4 identity matirx
• void glLoadMatrixfd(cost TYPE M)
• Sets the current matrix to 4X4 matrix specified
  by M
• Note current matrix ? Top most matrix of the
  current matrix stack

glLoadIdentity
glLoadMatrix(M)
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General Transformation Commands

• Concatenate Current Matrix
• void glMultMatrix(const TYPE M)
• Multiplies current matrix C, by M. i.e. C CM
• Caveat OpenGL matrices are stored in column
  major order.
• Best use utility function glTranslate, glRotate,
  glScale for common transformation tasks.

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Transformations and OpenGL

• Each time an OpenGL transformation M is called
  the current MODELVIEW matrix C is altered

glTranslatef(1.5, 0.0, 0.0) glRotatef(45.0, 0.0,
0.0, 1.0)
Note v is any vertex placed in rendering
pipeline v is the transformed vertex from v.
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Sample Instance Transformation

• glMatrixMode(GL_MODELVIEW)
• glLoadIdentity()
• glTranslatef(...)
• glRotatef(...)
• glScalef(...)
• gluCylinder(...)

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Thinking About Transformations

• There is a World Coordinate System where
• All objects are defined
• Transformations are in World Coordinate space

Two Different Views

• As a Global System
• Objects moves but coordinates stay the same
• Think of transformation in reverse order as they
  appear in code

• As a Local System
• Objects moves and coordinates move with it
• Think of transformation in same order as they
  appear in code

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Order of Transformation TR
glLoadIdentity() glMultiMatrixf(
T) glMultiMatrixf( R) draw_ the_ object( v) v
ITRv
Effect is same, but perception is different
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Order of Transformation RT
glLoadIdentity() glMultiMatrixf(
R) glMultiMatrixf( T) draw_ the_ object( v) v
ITRv
Effect is same, but perception is different
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Hierarchical Modeling

• Many graphical objects are structured
• Exploit structure for
• Efficient rendering
• Concise specification of model parameters
• Physical realism
• Often we need several instances of an object
• Wheels of a car
• Arms or legs of a figure
• Chess pieces
• Encapsulate basic object in a function
• Object instances are created in standard form
• Apply transformations to different instances
• Typical order scaling, rotation, translation

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OpenGL Hierarchical Model
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Scene Graph

• A scene graph is a hierarchical representation of
  a scene
• We will use trees for representing hierarchical
  objects such that
• Nodes represent parts of an object
• Topology is maintained using parent-child
  relationship
• Edges represent transformations that applies to a
  part and all the subparts connected to that part

Scene
typedef struct treenode GLfloat m16 //
Transformation void (f) ( ) // Draw
function struct treenode sibling struct
treenode child treenode
Sun
Star X
Earth
Venus
Saturn
Moon
Ring
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Example - Torso

• Initializing transformation matrix for node
• treenode torso, head, ...
• / in init function /
• glLoadIdentity()
• glRotatef(...)
• glGetFloatv(GL_MODELVIEW_MATRIX, torso.m)
• Initializing pointers
• torso.f drawTorso
• torso.sibling NULL
• torso.child head

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Generic Traversal

• To render the hierarchy
• Traverse the scene graph depth-first
• Going down an edge
• push the top matrix onto the stack
• apply the edge's transformation(s)
• At each node, render with the top matrix
• Going up an edge
• pop the top matrix off the stack

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Generic Traversal Torso

• Recursive definition
• void traverse (treenode root)
• if (root NULL) return
• glPushMatrix()
• glMultMatrixf(root-gtm)
• root-gtf()
• if (root-gtchild ! NULL) traverse(root-gtchild)
  
• glPopMatrix()
• if (root-gtsibling ! NULL) traverse(root-gtsibli
  ng)
• C is really not the right language for this !!

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Viewing Transformation
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Viewing Pipeline Revisited
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Viewing Transformation in OpenGL

• To setup the modelview matrix, OpenGL provides
  the following function

gluLookAt( eyex, eyey, eyez, centerx, centery,
centerz, upx, upy, upz )
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Implementation
We want to construct an Orthogonal Frame such
that,
(1) its origin is the point eye
(2) its -z basis vector points towards the point
center
(3) the up vector projects to the up direction
(ve y-axis)
Let C (for camera) denote this frame. Clearly,
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Thank You