Transcript PowerPoint

CS 551 / 645:
Introductory Computer Graphics
Geometric Transforms
Administrivia
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Return assignment 1
Hand out assignment 2
Today’s reading material: FvD, Chapter 5
Recap: Rendering Pipeline (3-D)
Scene graph
Object geometry
Result:
Modeling
Transforms
• All vertices of scene in shared 3-D “world” coordinate system
Lighting
Calculations
• Vertices shaded according to lighting model
Viewing
Transform
• Scene vertices in 3-D “view” or “camera” coordinate system
Clipping
Projection
Transform
• Exactly those vertices & portions of polygons in view frustum
• 2-D screen coordinates of clipped vertices
Recap: Transformations
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Modeling transforms
– Size, place, scale, and rotate objects parts of the
model w.r.t. each other
– Object coordinates  world coordinates
Y
Y
X
Z
Z
X
Rigid-Body Transforms
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Goal: object coordinatesworld coordinates
Idea: use only transformations that preserve
the shape of the object
– Rigid-body or Euclidean transforms
– Includes rotation, translation, and scale
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To reiterate: we will represent points as
column vectors:
 x


( x, y , z )   y 
 z 
Translations
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For convenience we usually describe objects
in relation to their own coordinate system
We can translate or move points to a new
position by adding offsets to their coordinates:
 x'  x  tx 
 y '   y   ty 
     
 z '   z  tz 
– Note that this translates all points uniformly
Scaling
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Scaling a coordinate means multiplying each
of its components by a scalar
Uniform scaling means this scalar is the
same for all components:
2
Scaling
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Non-uniform scaling: different scalars per
component:
X  2,
Y  0.5
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How can we represent this in matrix form?
Scaling
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Scaling operation:
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Or, in matrix form:
 x' ax 
 y '  by 
   
 z '   cz 
 x '   a 0 0  x 
 y'  0 b 0  y 
  
 
 z '  0 0 c   z 
scaling matrix
2-D Rotation
(x’, y’)
(x, y)
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x’ = x cos() - y sin()
y’ = x sin() + y cos()
2-D Rotation
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This is easy to capture in matrix form:
 x' cos  sin   x 
 y '   sin  cos   y 
  
 
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3-D is more complicated
– Need to specify an axis of rotation
– Simple cases: rotation about X, Y, Z axes
3-D Rotation
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What does the 3-D rotation matrix look like
for a rotation about the Z-axis?
– Build it coordinate-by-coordinate
 x' cos()  sin( ) 0  x 
 y '   sin( ) cos() 0  y 
  
 
 z '   0
0
1  z 
3-D Rotation
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What does the 3-D rotation matrix look like
for a rotation about the Y-axis?
– Build it coordinate-by-coordinate
 x'  cos() 0 sin( )   x 
 y '   0



1
0
y
  
 
 z '   sin( ) 0 cos()  z 
3-D Rotation
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What does the 3-D rotation matrix look like
for a rotation about the X-axis?
– Build it coordinate-by-coordinate
0
0  x
 x' 1
 y '  0 cos()  sin( )  y 
  
 
 z '  0 sin( ) cos()   z 
3-D Rotation
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General rotations in 3-D require rotating
about an arbitrary axis of rotation
Deriving the rotation matrix for such a
rotation directly is a good exercise in linear
algebra
Standard approach: express general rotation
as composition of canonical rotations
– Rotations about X, Y, Z
Composing Canonical Rotations
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Goal: rotate about arbitrary vector A by 
– Idea: we know how to rotate about X,Y,Z
So, rotate about Y by  until A lies in the YZ plane
Then rotate about X by  until A coincides with +Z
Then rotate about Z by 
Then reverse the rotation about X (by -)
Then reverse the rotation about Y (by -)
Composing Canonical Rotations
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First: rotating about Y by  until A lies in YZ
– Draw it…
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How exactly do we calculate ?
– Project A onto XZ plane
– Find angle  to X:
 = -(90° - ) =  - 90 °
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Second: rotating about X by  until A lies on Z
How do we calculate ?
Composing Canonical Rotations
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Why are we slogging through all this tedium?
A: Because you’ll have to do it on the test
3-D Rotation Matrices
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So an arbitrary rotation about A composites
several canonical rotations together
We can express each rotation as a matrix
Compositing transforms == multiplying
matrices
Thus we can express the final rotation as the
product of canonical rotation matrices
Thus we can express the final rotation with a
single matrix!
Compositing Matrices
So we have the following matrices:
p: The point to be rotated about A by 
Ry : Rotate about Y by 
Rx  : Rotate about X by 
Rz : Rotate about Z by 
Rx  -1: Undo rotation about X by 
Ry-1 : Undo rotation about Y by 
 In what order should we multiply them?
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Compositing Matrices
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Short answer: the transformations, in order,
are written from right to left
– In other words, the first matrix to affect the vector
goes next to the vector, the second next to the
first, etc.
So in our case:
p’ = Ry-1 Rx  -1 Rz Rx  Ry p
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Rotation Matrices
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Notice these two matrices:
Rx  : Rotate about X by 
Rx  -1: Undo rotation about X by 
How can we calculate Rx  -1?
Rotation Matrices
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Notice these two matrices:
Rx  : Rotate about X by 
Rx  -1: Undo rotation about X by 
How can we calculate Rx  -1?
– Obvious answer: calculate Rx (-)
– Clever answer: exploit fact that rotation matrices
are orthonormal
Rotation Matrices
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Notice these two matrices:
Rx  : Rotate about X by 
Rx  -1: Undo rotation about X by 
How can we calculate Rx  -1?
– Obvious answer: calculate Rx (-)
– Clever answer: exploit fact that rotation matrices
are orthonormal
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What is an orthonormal matrix?
What property are we talking about?
Rotation Matrices
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Orthonormal matrix:
– orthogonal (columns/rows linearly independent)
– Columns/rows length of 1
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The inverse of an orthogonal matrix is just its
transpose:
a b
d e

 h i
1
c
a b


f   d e
 h i
j 
T
c
a


f   b
 c
j 
d
e
f
h

i
j 
Translation Matrices?
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We can composite scale matrices just as we
did rotation matrices
But how to represent translation as a matrix?
Answer: with homogeneous coordinates
Homogeneous Coordinates
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Homogeneous coordinates: represent
coordinates in 3 dimensions with a 4-vector
 x / w  x 
 y / w  y 
 
( x, y , z )  
 z / w  z 

  
 1   w
(Note that typically w = 1 in object coordinates)
Homogeneous Coordinates
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Homogeneous coordinates seem unintuitive,
but they make graphics operations much
easier
Our transformation matrices are now 4x4:
0
0
1
0 cos()  sin( )
Rx  
0 sin( ) cos()

0
0
0
0
0
0

1
Homogeneous Coordinates
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Homogeneous coordinates seem unintuitive,
but they make graphics operations much
easier
Our transformation matrices are now 4x4:
 cos()
 0
Ry  
 sin( )

 0
0 sin( ) 0
1
0
0
0 cos() 0

0
0
1
Homogeneous Coordinates
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Homogeneous coordinates seem unintuitive,
but they make graphics operations much
easier
Our transformation matrices are now 4x4:
cos()  sin( )
 sin( ) cos()
Rz  
 0
0

0
 0
0 0
0 0
1 0

0 1
Homogeneous Coordinates
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Homogeneous coordinates seem unintuitive,
but they make graphics operations much
easier
Our transformation matrices are now 4x4:
 Sx 0
 0 Sy
S
0 0

0 0
0 0
0 0
Sz 0 

0 1
Homogeneous Coordinates
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How can we represent translation as a
4x4 matrix?
A: Using the rightmost column:
0
0
T
0

0
0 0 Tx 
0 0 Ty 
0 0 Tz 

0 0 1
Translation Matrices
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Now that we can represent translation as a
matrix, we can composite it with other
transformations
Ex: rotate 90° about X, then 10 units down Z:
 x'  1
 y '  0
 
 z '  0
  
 w' 0
0 0 0  1
0
0
1 0 0  0 cos(90)  sin( 90)
0 1 10 0 sin( 90) cos(90)

0 0 1  0
0
0
0  x 
0  y 
0  z 
 
1   w
Translation Matrices
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Now that we can represent translation as a
matrix, we can composite it with other
transformations
Ex: rotate 90° about X, then 10 units down Z:
 x'  1
 y '  0
 
 z '  0
  
 w' 0
0 0 0  1
1 0 0  0
0 1 10 0

0 0 1  0
0 0 0  x 
0  1 0  y 
1 0 0  z 
 
0 0 1   w
Translation Matrices
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Now that we can represent translation as a
matrix, we can composite it with other
transformations
Ex: rotate 90° about X, then 10 units down Z:
 x'  1
 y '  0
 
 z '  0
  
 w' 0
0 0 0  x 
0  1 0   y 
1 0 10  z 
 
0 0 1   w
Translation Matrices
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Now that we can represent translation as a
matrix, we can composite it with other
transformations
Ex: rotate 90° about X, then 10 units down Z:
 x'   x 
 y'   z 
 

 z '   y  10
  

 w'  w 
Transformation Commutativity
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Is matrix multiplication, in general,
commutative? Does AB = BA?
What about rotation, scaling, and translation
matrices?
– Does RxRy = RyRx?
– Does RAS = SRA ?
– Does RAT = TRA ?
More On Homogeneous Coords
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What effect does the following matrix have?
 x'  1
 y '  0
 
 z '  0
  
 w' 0
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0  x 



0  y 
0  z 
 
0 0 10  w
0 0
1 0
0 1
Conceptually, the fourth coordinate w is a bit
like a scale factor
Homogenous Coordinates
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In particular, increasing w makes things
smaller
We think of homogenous coordinates as
defining a projective space
– Increasing w  “getting further away”
– Points with w = 0 are, in this sense, projected on
the plane at infinity
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Will come in handy for projection matrices
Perspective Projection
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We talked about geometric transforms,
focusing on modeling transforms
– Ex: translation, rotation, scale, gluLookAt()
– These are encapsulated in the OpenGL
modelview matrix
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Can also express perspective projection (and
other projections) as a matrix
Next few slides: representing perspective
projection with the projection matrix
Perspective Projection
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In the real world, objects exhibit perspective
foreshortening: distant objects appear smaller
The basic situation:
Perspective Projection
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When we do 3-D graphics, we think of the
screen as a 2-D window onto the 3-D world:
How tall should
this bunny be?
Perspective Projection
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The geometry of the situation is that of
similar triangles. View from above:
View
plane
X
x’ = ?
(0,0,0)
Z
d
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P (x, y, z)
What is x’?
Perspective Projection
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Desired result for a point [x, y, z, 1]T projected
onto the view plane:
x' x
 ,
d z
dx
x
x' 

,
z
z d
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y' y

d z
dy
y
y' 

, zd
z
z d
What could a matrix look like to do this?
A Perspective Projection Matrix
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Answer:
1
0
Mperspective  
0

0
0
1
0
0
0
1
0 1d
0

0
0

0
A Perspective Projection Matrix
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Example:
 x  1
 y  0


 z  0

 
 z d  0
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0
1
0
0
0
1
0 1d
0  x 



0  y 
0  z 
 
0  1 
Or, in 3-D coordinates:
 x

,
z d

y
, d 
zd

Projection Matrices
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Now that we can express perspective
foreshortening as a matrix, we can composite
it onto our other matrices with the usual
matrix multiplication
End result: a single matrix encapsulating
modeling, viewing, and projection transforms
Matrix Operations In OpenGL
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Certain commands affect the current matrix
in OpenGL
– glMatrixMode() sets the current matrix
– glLoadIdentity() replaces the current matrix
with an identity matrix
– glTranslate() postmultiplies the current matrix
with a translation matrix
– gluPerspective() postmultiplies the current
matrix with a perspective projection matrix
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It is important that you understand the order
in which OpenGL concatenates matrices
Matrix Operations In OpenGL
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In OpenGL:
– Vertices are multiplied by the modelview matrix
– The resulting vertices are multiplied by the
projection matrix
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Example:
– Suppose you want to scale an object, translate it,
apply a lookat transformation, and view it under
perspective projection. What order should you
make calls?
Matrix Operations in OpenGL
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Problem: scale an object, translate it, apply a lookat
transformation, and view it under perspective
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A correct code fragment:
glMatrixMode(GL_PERSPECTIVE);
glLoadIdentity();
gluPerspective(…);
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
gluLookAt(…);
glTranslate(…);
glScale(…);
/* Draw the object... */
Matrix Operations in OpenGL
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Problem: scale an object, translate it, apply a lookat
transformation, and view it under perspective

An incorrect code fragment:
glMatrixMode(GL_PERSPECTIVE);
glLoadIdentity();
glTranslate(…);
glScale(…);
gluPerspective(…);
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
gluLookAt(…);
/* Draw the object... */