Transcript Introduction to Computer Graphics CS 445 / 645 Gimbal Lecture 10 Chapter 7: Transformations Overview Rotation representations • Euler • Axis-angle • Quaternion Camera Transformations • Projections.

Introduction to
Computer Graphics
CS 445 / 645
Gimbal
Lecture 10
Chapter 7: Transformations
Overview
Rotation representations
• Euler
• Axis-angle
• Quaternion
Camera Transformations
• Projections
Basic 3D Transformations
Rotate around Z axis:
 x' cos
 y '  sin 
 z'   0
 w   0
Rotate around Y axis:
 x '  cos
 y '  0
 
 z '   sin 
  
w   0
Rotate around X axis:
 x '  1
 y ' 0
 z '   0
 w  0
 sin 
cos 
0
0
0
cos 
sin 
0
0
1
0
0
0
0
1
0
0  x 
0  y 
0  z 
1  w
sin 
0
cos
0
0  x 
0  y 
0  z 
 
1  w 
0
 sin 
cos 
0
0  x 
0  y 
0  z 
1  w
3-D Rotation
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
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
First: rotating about Y by  until A lies in YZ
How exactly do we calculate ?
• Project A onto XZ plane (Throw away y-coordinate)
• Find angle  that rotates A to x-axis:
 = -(90° - ) =  - 90 °
Second: rotating about x-axis by  until A lies on z-axis
How do we calculate ?
Composing 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?
Compositing Matrices
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
Rotation Matrices
Notice these two matrices:
Rx  : Rotate about X by 
Rx  -1: Undo rotation about X by 
How can we calculate Rx  -1?
Rotation Matrices
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
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
What is an orthonormal matrix?
What property are we talking about?
Rotation Matrices
Orthonormal matrix:
• orthogonal (columns/rows linearly independent)
• normalized (columns/rows length of 1)
The inverse of an orthogonal matrix is just its
transpose:
a
d

 h
b
e
i
c

f
j 
1
a

 d
 h
b
e
i
T
c
a


f   b
 c
j 
d
e
f
h

i
j 
Representing 3 Rotational DOFs
3x3 Matrix (9 DOFs)
• Rows of matrix define orthogonal axes
Euler Angles (3 DOFs)
• Rot x + Rot y + Rot z
Axis-angle (4 DOFs)
• Axis of rotation + Rotation amount
Quaternion (4 DOFs)
• 4 dimensional complex numbers
Rotation Matrices
What did we start with?
• 3 rotations (roll, pitch, yaw)
– Three numbers
– Called Euler Angles
• Axis-angle
– Four numbers
And we end up with a 4x4 transformation matrix
• Where inner 3x3 describes rotation
Really, a whole matrix to represent three
numbers? Underconstrained?
9 DOFs must reduce to 3
Rows must be unit length (-3 DOFs)
Rows must be orthogonal (-3 DOFs)
Drifting matrices is very bad
• Numerical errors results when trying to gradually rotate matrix by
adding derivatives
• Resulting matrix may scale / shear
• Gram-Schmidt algorithm will re-orthogonalize your matrix
Difficult to interpolate between matrices
• Why would we do this?
Euler Angles
(x, y, z) = RzRyRx
• Rotate x degrees about x-axis
• Rotate y degrees about y-axis
• Rotate z degrees about z-axis
Axis order is not defined
• (y, z, x), (x, z, y), (z, y, x)…
are all legal
• Pick one
Euler Angles
Rotations not uniquely defined
• ex: (z, x, y) = (90, 45, 45) = (45, 0, -45)
takes positive x-axis to (1, 1, 1)
• Cartesian coordinates are independent of one another, but
Euler angles are not
Gimbal Lock
• Term derived from mechanical problem that arises in
gimbal mechanism that supports a compass or a gyro
A Gimbal
Hardware implementation of Euler angles (used for
mounting gyroscopes and globes)
Gimbal Lock
Gimbal Lock
Occurs when two axes are
aligned
Second and third rotations
have effect of transforming
earlier rotations
• ex: Rot x, Rot y, Rot z
– If Rot y = 90 degrees,
Rot z == -Rot x
Gimbal Lock
http://www.anticz.com/eularqua.htm
Interpolation
Interpolation between two Euler angles is not
unique
ex: (x, y, z) rotation
• (0, 0, 0) to (180, 0, 0) vs. (0, 0, 0) to (0, 180, 180)
• Interpolation about different axes are not
independent
Interpolation
Axis-angle Notation
Define an axis of rotation (x, y, z) and a rotation
about that axis, : R(, n)
4 degrees of freedom specify 3 rotational degrees
of freedom because axis of rotation is
constrained to be a unit vector
Axis-angle Rotation
Given
r – Vector in space to rotate
n – Unit-length axis in space about which to rotate
 – The amount about n to rotate
Solve
r’ – The rotated vector
n r
r’
Axis-angle Rotation
Step 1
• Compute rk an extended version of the rotation axis, n
• rk = (n ¢ r) n
rk
r’
r
Axis-angle Rotation
Compute r?
r? = r – (n ¢ r) n
r?
r
r’
Axis-angle Rotation
Compute v, a vector perpendicular to rk and r?
v = rk £ r?
Use v and r? and  to compute r’
v
cos() r? + sin() v

r?
Axis-angle Notation
rperp = r – (n.r) n

V = n x (r – (n.r) n) = n x r
Rr
r
rpar = (n.r) n
n
Rr = Rrpar + Rrperp
= Rrpar + (cos ) rperp + (sin ) V
=(n.r) n + cos (r – (n.r)n) + (sin ) n x r
= (cos )r + (1 – cos ) n (n.r) + (sin ) n x r
Axis-angle Notation
No easy way to determine how to concatenate
many axis-angle rotations that result in final
desired axis-angle rotation
No simple way to interpolate rotations
Quaternion
Remember complex numbers: a + ib
• Where i2 = -1
Invented by Sir William Hamilton (1843)
• Remember Hamiltonian path from Discrete II?
Quaternion:
• Q = a + bi + cj + dk
– Where i2 = j2 = k2 = -1 and ij = k and ji = -k
• Represented as: q = (s, v) = s + vxi + vyj + vzk
Quaternion
A quaternion is a 4-D unit vector q = [x y z w]
• It lies on the unit hypersphere x2 + y2 + z2 + w2 = 1
For rotation about (unit) axis v by angle 
• vector part = (sin /2) v = [x y z]
• scalar part = (cos /2)
=w
• (sin(/2) vx, sin(/2) vy, sin(/2) vz, cos (/2))
Only a unit quaternion encodes a rotation - normalize
Quaternion
Rotation matrix corresponding to a quaternion:
• [x y z w] = 1  2 y 2  2 z 2
2 xy  2wz
2 xz  2wy 


2
2
2
xy

2
wz
1

2
x

2
z
2
yz

2
wx


2
2
 2 xz  2wy
2
yz

2
wx
1

2
x

2
y


Quaternion Multiplication
• q1 * q2 = [v1, w1] * [v2, w2] = [(w1v2+w2v1+ (v1 x v2)), w1w2-v1.v2]
• quaternion * quaternion = quaternion
• this satisfies requirements for mathematical group
• Rotating object twice according to two different quaternions is equivalent
to one rotation according to product of two quaternions
Quaternion Example
X-roll of p
• (cos (p/2), sin (p/2) (1, 0, 0)) = (0, (1, 0, 0))
Y-roll 0f p
• (0, (0, 1, 0))
Z-roll of p
• (0, (0, 0, 1))
Ry (p) followed by Rz (p)
• (0, (0, 1, 0) times (0, (0, 0, 1)) = (0, (0, 1, 0) x (0, 0, 1)
= (0, (1, 0, 0))
Quaternion Interpolation
Biggest advantage of quaternions
• Interpolation
• Cannot linearly interpolate between two quaternions
because it would speed up in middle
• Instead, Spherical Linear Interpolation, slerp()
• Used by modern video games for third-person
perspective
• Why?
SLERP
Quaternion is a point on the 4-D unit sphere
•
interpolating rotations requires a unit quaternion at each step
– another point on the 4-D unit sphere
•
move with constant angular velocity along the great circle between two points
– A great circle is a section of a sphere that contains a diameter of the sphere
(Kern and Bland 1948, p. 87).
Any rotation is defined by 2 quaternions, so pick the shortest SLERP
To interpolate more than two points, solve a non-linear variational
constrained optimization
•
Ken Shoemake in SIGGRAPH ’85 (www.acm.org/dl)
Quaternion Interpolation
Quaternion (white) vs.
Euler (black)
interpolation
Left images are linear
interpolation
Right images are cubic
interpolation
Quaternion Code
http://www.gamasutra.com/features/programming/
19980703/quaternions_01.htm
• Registration required
Camera control code
• http://www.xmission.com/~nate/smooth.html
– File, gltb.c
– gltbMatrix and gltbMotion
3D
Rendering
Pipeline
(for direct illumination)
3D Geometric Primitives
Modeling
Transformation
Lighting
Transform into 3D world coordinate system
Illuminate according to lighting and reflectance
Viewing
Transformation
Transform into 3D camera coordinate system
Projection
Transformation
Transform into 2D screen coordinate system
Clipping
Scan
Conversion
Image
Clip primitives outside camera’s view
Draw pixels (includes texturing, hidden surface, ...)
Reminder: Homogeneous Coords
What effect does the following matrix have?
 x'  1
 y '  0
 
 z '  0
  
 w' 0
0
1
0
0
0 0  x 



0 0  y 
1 0  z 
 
0 10  w
Conceptually, the fourth coordinate w is a bit like a scale
factor
Homogenous Coordinates
Increasing w makes things smaller
We think of homogenous coordinates as defining
a projective space
• Increasing w  “getting further away”
Will come in handy for projection matrices
Projection Matrix
We talked about geometric transforms, focusing on
modeling transforms
• Ex: translation, rotation, scale, gluLookAt()
• These are encapsulated in the OpenGL modelview matrix
Projection is also represented as a matrix
Next few slides: representing orthographic and
perspective projection with the projection matrix
Taxonomy of Projections
FVFHP Figure 6.10
Taxonomy of Projections
Parallel Projection
Center of projection is at infinity
• Direction of projection (DOP) same for all points
View
Plane
DOP
Angel Figure 5.4
Orthographic Projections
DOP perpendicular to view plane
Front
Angel Figure 5.5
Top
Side
Oblique Projections
DOP not perpendicular to view plane
  45
Cavalier
o
(DOP  = 45 )
  63.4
Cabinet
o
(DOP  = 63.4 )
H&B Figure 12.24
Orthographic Projection
Simple Orthographic
Transformation
Original world units are preserved
• Pixel units are preferred
Orthographic: Screen Space
Transformation
left =10 m
right = 20 m
top=20 m
(max pixx, max pixy)
(height in pixels)
bottom=10 m
(0, 0)
(width in pixels)
Orthographic: Screen Space
Transformation
left, right, top, bottom refer to the viewing frustum in
modeling coordinates
width and height are in pixel units
This matrix scales and translates to accomplish the
transition in units