CG OpenGL vectors geometric & transformations-course 5

Vectors & Geometric
Transformations
Chen Jing-Fung (2006/11/17)
Assistant Research Fellow,
Digital Media Center,
National Taiwan Normal University

Video Processing Lab
臺灣師範大學數位媒體中心視訊處理研究室

National Taiwan Normal University
Ch5: Computer Graphics with OpenGL 3th, Hearn Baker

3D Graphics
• Goal
– Using mathematical method produce 2D images
which is described 3D
• Problems
– Modeling (environment)
• Construct a scene
• Hierarchical description of a complex object (several
parts composed) -> simpler parts
– Rendering
• Described how objects move around in an animation
sequence?
• Simply to view objects from another angle

Video Processing Lab 2

Coordinate Systems
• Coordinate systems are fundamental
in computer graphics
– Describe the locations of points in space
• Project from one coordinate system
to another
– Easier to understand and implement


One coordinate system
• Scalars (α), Point (P) & Vector (V)
y
P2: (x2, y2)
V2=[x2, y2] vx vy y
V=P2-P1=[x2- x1, y2-y1]=[vxy]
P1: (x1, y1) b
a+b
V1=[x1, y1] x
a x
Scalars (α)

α V=[αvij] Where is a-b & ab ??


Two vectors in one
coordinate system
• Scalar product or dot product V2

– V1.V2=|V1| |V2| cos Θ V1
• Commutative V1.V2= V2.V1 Θ

• Associative V1.(V2+V3)= V1.V2+V1.V3
• Vector product or cross product V1xV2
V2
– V1xV2=u|V1| |V2| sin Θ Θ
• Anti-commutative V1xV2= -V2xV1 u: unit vector V
1
• No associative V1x(V2xV3) ≠ (V1xV2)xV3
• Associative V1x(V2+V3)= (V1xV2)+(V1xV3)
u x u y uz
V1  V2  v1x v1y v1z
v 2 x v 2y v 2z

Two coordinate systems
• The point is represented by two different
coordinate systems
y
v
(u, v)

u
(4, 6)

x
– Maybe only one coordinate system can
represent the point by the easier way


Transformations (1)

• Transformation functions between
two Coordinate systems (rendering)
y v
(u, v)
u=x-3
v=y-4
u
(4, 6) x=u+3
y=v+4
x


Transformations (2)

• In the same coordinate system to
modify an object’s shape
y y
x’=x-2
y’=y-3
(4, 6)
(2, 3)
x=x’+2
y=y’+3
x x


2D affine
transformations
• Affine transformation
– Linear projected function x 
x
x '  axx x  axy y  t x linear
xy x  x  y
y '  ayx x  ayy y  t y y 

 x '  axx axy   x  t x 
or  y '   a     t 
ayy   y   y 
   yx


Composition of affine
transforms
• Simple transformations
– Translation
– Scaling
– Rotation
– Shear
– Reflection


2D Translation
• Translation equation P'  P  T y P’3

 x '   1 0   x  t x 
y

 y '   0 1  y   t  ty
      y x
x tx

– Translation is a rigid-body transformation that
moves objects without deformation.
– Delete the original polygon
• To cover the background color (& save it in different
array)


2D Scaling
• Scaling equation P'  S  P
y
 x '  s x 0   x  0  y

 y '   0     
sy   y  0 
   syy
x
x
sxx

– Uniform scaling: sx=sy sx>1, sy< 1

– Differential scaling: sx≠sy


2D Rotation (1)

• Single point rotation
– Pivot-point = origin
x '  r cos(   ), y '  r sin(   )
(x’,y’)
x  r cos( ), y  r sin( )
r (x,y)

x '  cos * x  sin * y
y '  sin * x  cos * y


2D Rotation (2)
• Rotation equation P  R( )  P '

 x '  cos  sin   x  0
 y '   sin    y   0 
cos     
   

– Now, graphics packages all follow the standard
column-vector convention
• OpenGL, Java, Matlab …
– Trans. and rotations are rigid-body transformations
that move object without deformation
• Each point on object is rotation through the same angle
• Must be defined the direction of the rotated angle
Rotation point is also called pivot point

Y-axis Shear

• Shear along y axis (what is x-axis
shear?)
 x '  1 0   x  0 
 y '   sh   
y
y
   y 1  y  0 

x
x


Y axis Reflection

 x '   1 0   x  0 
 y '   0 1  y   0
y
y
      

x
x

• What is the reflection of x-axis?


Basic transformations
• General form: P'  M2  P  M1
– Translation (shift): M1=identity matrix
– Rotation or scaling: M2=translation term (R(Θ))
or scaling fixed pixel (S(s*))
• The efficient approach
– To produce a sequence of transformations with
these equation
• Scale -> rotate(Θ) -> translate (linear)
• The final coordinate positions are obtained directly
from initial coordinates


Homogeneous coordinate
• General transformation equations:
– x’ = axx + axy + tx  x '  axx axy tx   x 
 y '   a ty   y 
y’ = ayx + ayy + ty 2D    yx ayy   
1  0
   0 1   1
  
• A standard technique is used to expand
the matrix 2D (x,y) -> 3D(x,y,z) h*x
– Homogenous coordinates: (xh, yh, h)
– Homogenous parameter: h h*y
• ‘h’ means the number of points in z-axis
– Simply to set h=1


Basic Matrix3x3
• Translation matrix 1 0 tx 
P'  P  T(t* )  P'  T(t* )  P 0 1 t 
 y
0 0 1 
 
• Scaling matrix
s x 0 0
P'  S(s* )  P 0 sy 0
 
0
 0 1

• Rotation matrix
cos  sin 0
P'  R( )  P  sin cos  0
 
 0
 0 1



Arbitrary point’s
rotation (1)
• Single point rotation
– Pivot-point (xr,yr) ! = original point
1. translate
• How to find solution u  x  xr
– !! Coordinate transformation v  y  yr
2. rotate
u '  r cos(   ),v '  r sin(   ) u '  x ' x r
(u’,v’)
u  r cos( ),v  r sin( ) v '  y ' y r
r (u,v)

x '  xr  ( x  xr )cos  ( y  y r )sin
(xr,yr)
y '  y r  ( x  xr )sin  ( y  y r )cos


Arbitrary point’s
rotation (2)

• General 2D point rotation (or scaling)
– (xr, yr) & (xr, yr) ≠Origin
• Translate origin coordinate -> the point
position T( xr , y r )
• Rotate (or scaling) the object about the
coordinate origin R( ) or S(sx , sy )
• Translate the point returned to its original
position T( xr , y r )


Pivot-point rotation
composite matrix
T( xr , y r )

R( ) T( xr , y r )

T( xc , yc )  R( )  T( xc , yc )  R( xc , y c , )
 1 0 X c  cos  sin 0   1 0 X c 
0 1 Y    sin cos 0   0 1 Y  x '  xr  ( x  xr )cos  ( y  y r )sin
 c     c  y '  y  ( x  x )sin  ( y  y )cos 
r r r
0 0 1   0
   0 1 0 0 0 
  
cos  sin xc (1  cos )  y c sin 
  sin cos y c (1  cos )  xc sin 
 
 0
 0 1 


Scaling an Object not at
the Origin
• What case happens?
– Apply the scaling transformation to an
object not at the origin?
• Based on the rotating about a point
composition, what should you do to
resize an object about its own center?

T( xc , yc )  S(sx , sy )  T(xc , y c )  S( xc , y c , sx , sy )


Back to Rotation About a Pt
• R (rotation matrix) and p (Pivot-point)
describe how to rotate
– Translation Origin to the position:
x  x  p
– Rotation:
x  Rx  R(x  p)  Rx  Rp
– Translate back:
x  x  p  Rx  Rp  p
• The composite transformation involves the
rotation matrix.
T( xnc , y nc )  R( )  T( xnc , y nc )

Matrix concatenation
properties
• What is matrix concatenation?
M3  M2  M1  (M3  M2 )  M1  M3  (M2  M1 )
– Multiplication of matrices is associative
• Premultiplying (left-to-right) = ??
Postmultiplying (right-to-left)
– Transformation products not be
commutative M2  M1  M1  M2


3D transformations

• Homogeneous coordinates  x '  a d g tx   x 
 y ' b e h ty  y 
– 4x4 matrices    
 z '  c f u tz  z 
     
 1  0 0 0 1   1

• Specification of translation, rotation,
scaling and other matrices in OpenGL
– glTranslate(), glRotate(), glScale(),
glMultMatrix()


3D translation & scaling

• 3D Translation  x ' 1 0 0 tx   x 
 y '  0 1 0 ty  y 
   
 z '  0 0 1 tz  z 
     
 1  0 0 0 1   1

• 3D Scaling
 x '  s x 0 0 0  x 
 y '  0 sy 0 0  y 
   
z '   0 0 sz 0 z 
     
1 0 0 0 1  1 


3D z-Axis Rotation

• 2D extend along z-axis y axis

– (2D->3D)  X '  cos
 ' 
 sin 0 X 
 Y    sin cos  0  Y 
  
 Z'   0 1  Z 
   0   

P  R z ( )  P
' x axis
z axis

 X '  cos   sin 0 0 X  counterclockwise
 ' 
 Y    sin cos  0 0 Y 
 
Z'   0 0 1 0 Z 
     
1   0
  0 0 1 1 

y-axis & x-axis?

3D Rotation of arbitraryy axis
y
y Step 2
Step 1 P’2
P2
P’1
P’1 P”2
P1 x
x z
x z P’2 rotate onto z-axis
z Initial position
y P1 translate to the Origin
y
Step 3 Step 4 y Step 5 P2
P’1 P’2
P”2 P1
x P’1
z x
z
Rotation the Object x Translate the rotation
z
around z-axis Rotate the axis to its axis to its Original
Original Orientation position
R( )  T  R  T 1
R  R 1( )  R 1(  )  R z ( )  R y (  )  R x ( )
x y

Problems with Rotation
Matrices
• Specifying a rotation really only requires 3
numbers in three Cartesian coordinates
– 2 numbers to show a unit vector
– Third number to show the rotation angle
• Rotation matrix has a large amount of
redundancy
– Orthonormal constraints reduce degrees of
freedom back down to 3
– Keeping a matrix orthonormal is difficult when
transformations are combined


Alternative
Representations
• Specify the axis and the angle (OpenGL method)
– Hard to compose multiple rotations
• Specify the axis, scaled by the angle
– Only 3 numbers, but hard to compose
• Euler angles:
– First, how much to rotate about X
– Second, how much to rotate about Y
– Final, how much to rotate about Z
• Hard to think about, and hard to compose
• Quaternions


Quaternions

• 4-vector related to axis and angle,
unit magnitude
– Rotation about axis (x,y,z) by angles θ:
 x '  cos   sin 0 0  x 
 y '   sin cos  0 0  y 
   
z '   0 0 1 0 z 
     
1   0 0 0 1 1 

• Easy to compose
• Easy to find rotation matrix


Transformation in
OpenGL
• Transformation pipeline & matrices
– Current Transformation Matrix (CTM)
– CTM operations
– CTM in OpenGL
– OpenGL matrices


Transformation pipeline &
matrices
object eye

Projection
Modelview Matrix
vertex
matrix

modelview projection
modelview

• OpenGL matrices have three types
– Model-View (GL_MODEL_VIEW)
– Projection (GL_PROJECTION)
– Texture (GL_TEXTURE) (ignore for now)


Current Transformation
Matrix (CTM)
• CTM is a 4x4 homogeneous
coordinate matrix
– It can be altered by a set of function
– It is defined in the user program
– and loaded into a transformation unit
C Current matrix

P P’=CP
vertices CTM vertices


CTM operations P
Current matrix
C
P’=CP
CTM
vertices vertices
• CTM can be altered by loading new matrix
or by postmultiply matrix
– Load form
glLoadIdentity();
• identity matrix: C←I
T: glTranslatef(dx, dy, dz);
• an arbitrary matrix: C←M
R: glRotatef(angle, vx, vy, vz);
• translation matrix: C←T … S: glScalef(sx, sy, sz);
– Postmultiply form glMultMatrixf( );
• an arbitrary matrix: C←CM
User input matrix
• a translation matrix: C←CT
• a rotation matrix: C←CR …


Example by point rotation
• Rotation with an arbitrary point
– Order of transformations in OpenGL
(one step = one function call)
Initial • Loading an identity matrix: C←I
• Translation Origin to the position: C← CT
• Rotation: C← CR
• Translate back: C← CT-1
– Result: C= TRT-1


CTM in OpenGL
• In OpenGL, CTM has the model-view
matrix and the projection matrix
CTM

Modelview Projection
vertices vertices
matrix matrix

Geometric transformations glMatrixMode routine

– Manipulate those matrices by concatenation
and start from first setting matrix


OpenGL matrices (1)

• Current matrix
glMatrixMode (GL_MODELVIEW|GL_PROJECTION)
• Arbitrary matrix
– Load 16-elements array glLoadMatrix* (elems);
• A suffix code: f or d
• The elements must be specified in column order
– First list 4-elements in first-column
– …
– Finally the fourth column
– Stack & store the Matrix
glPushMatrix ();
glPopMatrix ();

OpenGL matrices (2)
• Multiple by two arbitrary matrices
C<-M2M1
glLoadIdentity();
glMultMatrixf(elemsM2);
glMultMatrixf(elemsM1);

• Access matrices by query functions
glGetIntegerv
glGetFloatv
glGetBooleanv
glGetdoublev
glIsEnabled…

Summery

• Rotation related with axis and the origin
– Use the same trick as in 2D:
• Translate origin to the position
• Rotate
• translate back again
• Rotation is not commutative
– Rotation order matters
– Experiment to convince by yourself


Transformation trick

• Rotation and Translation are the
rigid-body transformations
– Do not change lengths, sizes or angles,
so a body does not deform
• Scale, shear… extend naturally
transformation from 2D to 3D


Triangle’s rotation at
arbitrary point
• vertices tri = {{50.0, 25.0}, {150.0, 40.0}, {100.0,
100.0}}; //set object’s vertices
• Centpt; //find center point to describe the
triangle
• glLoadIdentity();
• glTranslatef(); //translate the center point
• glRotatef(angle, vx,vy,vz); //rotate the center
point, axis=(vx,vy,vz), angle: user define
• glTranslatef(); //translate return


Middle project

• Make some visual components by yourself
– more than three object’s from HW1 & HW2
• Practice each one composition (rotation, scaling,
translate, shear and reflection)
– Note: original & new
• Practice combining two or three compositions
• Team work (2~3)


Reference
• http://www.cs.wisc.edu/~schenney/
• http://graphics.csie.ntu.edu.tw/~robi
n/courses/3dcg06/
• http://www.cse.psu.edu/~cg418/
• http://groups.csail.mit.edu/graphics/
classes/6.837


Inverse matrix
• Identify matrix (Inxn) 1 0 0
I I3 x 3  0 1 0 
– MM-1=I,  
1
M 
M  0 0 1
 
– Inverse matrix (M-1 )
1 0 tx   1 0 t x 
T  0 1 t y 
  T 1  0 1 t y 
 
0 0 1 
  0 0 1 
 
cos  sin 0  cos sin 0
R   sin
 cos 0
 R 1    sin
 cos 0   RT

 0
 0 1
  0
 0 1


2D Reflection
Reflection line y=0 Reflection line x=0 Reflection line y=-x
y y y

x x x

• Transformation matrix
1 0 0  1 0 0   0 1 0 
0 1 0   0 1 0  1 0 0 
     
 0 0 1
   0 0 1  0 0 1
   


CG OpenGL vectors geometric & transformations-course 5

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