CS 430/536
Computer Graphics I
Outline
3D Transformations
World Window to Viewport Transformation
Week 2, Lecture 4
•  World window to viewport transformation
•  3D transformations
•  Coordinate system transformation
David Breen, William Regli and Maxim Peysakhov
Geometric and Intelligent Computing Laboratory
Department of Computer Science
Drexel University
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http://gicl.cs.drexel.edu
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The Window-to-Viewport
Transformation
Window-to-Viewport
Transformation
•  Problem: Screen windows cannot display the whole
world (window management)
•  How to transform and clip:
Objects to Windows to Screen
•
Given a window
and a viewport,
what is the
transformation from
WCS to VPCS?
Three steps:
•  Translate
•  Scale
•  Translate
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1994 Foley/VanDam/Finer/Huges/Phillips ICG
Pics/Math courtesy of Dave Mount @ UMD-CP
Transforming World
Coordinates to Viewports
•
Clipping to the Viewport
3 steps
•  Viewport size may
not be big enough
for everything
•  Display only the
pixels inside the
viewport
1.  Translate
2.  Scale
3.  Translate
Overall Transformation:
P'= M wv P
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1994 Foley/VanDam/Finer/Huges/Phillips ICG
•  Transform lines in world
•  Then clip in world
• Transform to image
•  Then draw
•  Do not transform pixels
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1994 Foley/VanDam/Finer/Huges/Phillips ICG
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Another Example
3D Transformations
•  Scan-converted
–  Lines
–  Polygons
–  Text
–  Fill regions
•  Clip
regions
for display
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Representation of 3D
Transformations
3D Homogenous Coordinates
•  Z axis represents depth
•  Right Handed System
•  Homogenous
coordinates for 2D
space requires 3D
vectors & matrices
•  Homogenous
coordinates for 3D
space requires 4D
vectors & matrices
•  [x,y,z,w]
–  When looking down at the
origin, positive rotation is CCW
•  Left Handed System
–  When looking down , positive
rotation is in CW
–  More natural interpretation for
displays, big z means far
(into screen)
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1994 Foley/VanDam/Finer/Huges/Phillips ICG
3D Transformations:
Scale & Translate
3D Transformations:
Rotation
•  Scale
•  One rotation for
each world
coordinate axis
–  Parameters for each
axis direction
•  Translation
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Rotation Around an
Arbitrary Axis
Rotation Around an
Arbitrary Axis
The image cannot be displayed. Your computer may not have enough
memory to open the image, or the image may have been corrupted. Restart
your computer, and then open the file again. If the red x still appears, you
may have to delete the image and then insert it again.
•  Rotate a point P
around axis n (x,y,z)
by angle θ
# tx 2 + c
%
txy " sz
R =%
%txz + sy
%
$ 0
!
txy + sz txz " sy
ty 2 + c tyz + sx
tyz " sx tz 2 + c
0
0
•  c = cos(θ)
•  s = sin(θ)
•  t = (1 - c)
•  Also can be expressed as the
Rodrigues Formula
0&
(
0(
0(
(
1'
Graphics Gems I, p. 466 & 498
Prot = P cos(" ) + (n # P)sin(" ) + n(n $ P)(1% cos(" ))
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Improved Rotations
•  Euler Angles have problems
–  How to interpolate keyframes?
–  Angles aren t independent
–  Interpolation can create Gimble Lock, i.e.
loss of a degree of freedom when axes
align
•  Solution: Quaternions!
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!
p = (0, x )
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What about interpolating
multiple keyframes?
•  Shoemake suggests using Bezier
curves on the sphere
•  Offers a variation of the De Casteljau
algorithm using slerp and quaternion
control points
•  See K. Shoemake, Animating rotation
with quaternion curves , Proc.
SIGGRAPH 85
A & B are quaternions
slerp – Spherical linear interpolation
Need to take equals steps on the sphere
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3D Transformations:
Shear
3D Transformations:
Reflect & Shear
" X'% " 1
$ ' $ x
$ Y ' ' = $ Shy
$ Z' ' $ Shzx
$ ' $
#1& # 0
•  Reflection:
about x-y plane
Shxy
1
Shxz
Shyz
Shzy
0
1
0
0% " X %
' $ '
0' $ Y '
(
0' $ Z '
' $ '
1& # 1 &
X ' = X + ShxyY + Shxz Z
•  Shear:
(function of z)
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Y ' = Shyx X +Y + Shyz Z
Z ' = Shzx X + ShzyY + Z
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Example: Composition of 3D
Transformations
Example
•  Goal: Transform P1P2 and P1P3
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Pics/Math courtesy of Dave Mount @ UMD-CP
1994 Foley/VanDam/Finer/Huges/Phillips ICG
Example (Cont.)
•
Process
1.
2.
3.
4.
Translate P1 to (0,0,0)
Rotate about y
Rotate about x
Rotate about z
Final Result
•  What we ve really
done is transform
the local coordinate
system Rx, Ry, Rz
to align with the
origin x,y,z
(1)
(2-3)
(4)
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Example 2: Composition of 3D
Transformations
•  Airplane defined in x,y,z
•  Problem: want to point it
in Dir of Flight (DOF)
centered at point P
•  Note: DOF is a vector
•  Process:
1994 Foley/VanDam/Finer/Huges/Phillips ICG
Example 2 (cont.)
•  Zp axis to be DOF
•  Xp axis to be a horizontal
vector perpendicular to
DOF
–  y x DOF
•  Yp, vector perpendicular to
both Zp and Xp (i.e.Zp x Xp)
–  Rotate plane
–  Move to P
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1994 Foley/VanDam/Finer/Huges/Phillips ICG
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Transformations to Change
Coordinate Systems
•  Issue: the world has many different relative
frames of reference
•  How do we transform among them?
•  Example: CAD Assemblies & Animation Models
Transformations to Change
Coordinate Systems
•  4 coordinate systems
1 point P
M 1!2 = T (4,2)
M 2!3 = T (2,3) • S (0.5,0.5)
M 3!4 = T (6.7,1.8) • R(45! )
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Coordinate System Example (1)
•  Transformation
M 5"1 = M 5"4 • M 4"3 • M 3"2 • M 2"1
Composition:
•  Translate the House
to the origin
M1"2 = T(x1, y1 )
M 2"1 = ( M1"2 )
!
#1
= T(#x1,#y1 )
!
Coordinate System Example (2)
P1
The matrix Mij that maps points from coordinate system j
to i is the inverse of the matrix Mji that maps points from
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coordinate system j to coordinate system i.
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1994 Foley/VanDam/Finer/Huges/Phillips ICG
World Coordinates and
Local Coordinates
•  To move the tricycle,
we need to know how
all of its parts relate to
the WCS
•  Example: front wheel
rotates on the ground
wrt the front wheel s z
axis:
P (wo) = T("r,0,0) # Rz (" ) # P (wh )
Coordinates of P in
wheel coordinate
system:
P'(wh ) = Rz (" ) # P (wh )
1994 Foley/VanDam/Finer/Huges/Phillips ICG
Questions about Homework 1?
•  Go to web site
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