Post on 30-Dec-2015
CSE 681
Review:Transformations
CSE 681
Transformations• Modeling transformations
• build complex models by positioning (transforming) simple components relative to each other
• Viewing transformations
• placing virtual camera in the world
• transformation from world coordinates to camera coordinates
• Perspective projection of 3D coordinates to 2D
• Animation
• vary transformations over time to create motion
CSE 681
Transformations - Modeling
world
CSE 681
Transformations - Viewing
WORLD
OBJECTCAMERA
CSE 681
Modeling Transformations
Transform objects/points Transform coordinate system
CSE 681
Affine Transformations
Transform P = (x, y, z) to Q = (x’, y’, z’) by M.
Affine transformation:
34333231
24232221
14131211
'
'
'
mzmymxmz
mzmymxmy
mzmymxmx
CSE 681
Translation
Translation by (tx, ty, tz):
(tx, ty, tz)z
y
x
tzz
tyy
txx
'
'
'
CSE 681
Scaling
Scaling by (sx, sy, sz):
Uniform v. non-uniform scaling
zsz
ysy
xsx
z
y
x
'
'
'
CSE 681
Rotation
Rotation counter-clockwise by angle around the z-axis:
x’ = x cos() – y sin()
y’ = x sin() + y cos()
z’ = z
Proof:x = r cos()y = r sin()
x’ = r cos( + ) = r cos() cos() – r sin() sin() = x cos() – y sin()y’ = r sin( + ) = r cos() sin() + r sin() cos() = x sin() + y cos()
xz
y
r
r
CSE 681
Rotation around x-axis
Rotation counter-clockwise by angle around the x-axis:
x’ = x
y’ = y cos() – z sin()
z’ = y sin() + z cos()
yx
z
CSE 681
Rotation around y-axis
Rotation counter-clockwise by angle around the y-axis:
y’ = y
z’ = z cos() – x sin()
x’ = z sin() + x cos()
Or
x’ = x cos() + z sin()
y’ = y
z’ = – x sin() + z cos()zy
x
CSE 681
Matrix Multiplication
Rotation counter-clockwise by angle around the z-axis:x’ = x cos() – y sin() y’ = x sin() + y cos() z’ = z
.
00
00
00
'
'
'
z
y
x
s
s
s
z
y
x
z
y
x
x '
y '
z'
cos() sin() 0
sin() cos() 0
0 0 1
x
y
z
.
x '
y '
z'
?
x
y
z
.
Translation by (tx, ty, tz):
x’ = x + tx
y’ = y + ty
z’ = z + tz
Scaling by (sx, sy, sz):
x’ = sx x
y’ = sy y
z’ = sz z
CSE 681
Homogeneous Coordinates
Scaling by (sx, sy, sz):
x’ = sx x
y’ = sy y
z’ = sz z
.
11000
100
010
001
1
'
'
'
z
y
x
t
t
t
z
y
x
z
y
x
.
11000
000
000
000
1
'
'
'
z
y
x
s
s
s
z
y
x
z
y
x
Represent P by (x, y, z, 1) and Q by (x’, y’, z’, 1).
(Homogeneous coordinates.)
Translation by (tx, ty, tz):
x’ = x + tx
y’ = y + ty
z’ = z + tz
CSE 681
Rotation Matrices
Rotation counter-clockwise by angle around the x-axis:x’ = xy’ = y cos() – z sin() z’ = y sin() + z cos()
x '
y '
z'
1
cos() sin() 0 0
sin() cos() 0 0
0 0 1 0
0 0 0 1
x
y
z
1
.
.
11000
0)cos()sin(0
0)sin()cos(0
0001
1
'
'
'
z
y
x
z
y
x
.
11000
0)cos(0)sin(
0010
0)sin(0)cos(
1
'
'
'
z
y
x
z
y
x
Rotation counter-clockwise by angle around the z-axis:x’ = x cos() – y sin() y’ = x sin() + y cos() z’ = z
Rotation counter-clockwise by angle around the y-axis:
x’ = x cos() + z sin()
y’ = y
z’ = – x sin() + z cos()
CSE 681
Affine Transformation Matrix
Affine transformation:
x’ = m11 x + m12 y + m13 z + m14
y’ = m21 x + m22 y + m23 z + m24
z’ = m31 x + m32 y + m33 z + m34
Transformation Matrix:.
110001
'
'
'
34333231
24232221
14131211
z
y
x
mmmm
mmmm
mmmm
z
y
x
CSE 681
Shearing
Shear along the x-axis:
x’ = x + hy
y’ = y
z’ = z
xz
y
.
11000
0100
0010
001
1
'
'
'
z
y
xh
z
y
x
CSE 681
Reflection
Reflection across the x-axis:
x’ = (-1) x
y’ = y
z’ = zxz
y
Reflection is a special case of scaling!
.
11000
0100
0010
0001
1
'
'
'
z
y
x
z
y
x
CSE 681
Elementary Transformations
• Translation
• Rotation
• Scaling
• Shear
What about inverses?
CSE 681
Inverse Transformations
Translation
Rotation
Scale
Shear
CSE 681
Compose Transformations
xz
y• Scale by (2, 1, 1)
• Translate by (20, 5, 0)
• Rotate by 30° counter-clockwise around the z-axis
• Translate by (0, -50, 0)
CSE 681
Compose Transformation Matrices
xz
y• Scale by (2, 1, 1);
.
11000
0100
0010
0002
z
y
x
1
'
'
'
z
y
xScale
• Translate by (20, 5, 0);
1000
0100
5010
20001Translate
• Rotate by 30° counter-clockwise around the z-axis;
1000
0100
0086.05.0
005.086.0Rotate
• Translate by (0, -50, 0).
1000
0100
50010
0001Translate
CSE 681
Compose Transformation Matrices
xz
y• Scale by (2, 1, 1);
.
11000
0100
3.44088.00.1
7.1405.072.1
1
'
'
'
z
y
x
z
y
x
• Translate by (20, 5, 0);
• Rotate by 30° counter-clockwise around the z-axis;
• Translate by (0, -50, 0).
CSE 681
Order Matters!
Transformations are not necessarily commutative!
xz
y
• Translate by (20, 0, 0).
• Rotate by 30.
xz
y
• Rotate by 30.
• Translate by (20, 0, 0).
CSE 681
Order of Transformation Matrices
Apply transformation matrices from right to left.
• Translate by (20, 0, 0).
• Rotate by 30.
• Rotate by 30.
• Translate by (20, 0, 0)..
11000
0100
0086.05.0
005.086.0
1000
0100
0010
20001
1
'
'
'
z
y
x
z
y
x
.
11000
0100
0010
20001
1000
0100
0086.05.0
005.086.0
1
'
'
'
z
y
x
z
y
x
CSE 681
• Translation and translation?
• Scaling and scaling?
• Rotation and rotation?
• Translation and scaling?
• Translation and rotation?
• Scaling and rotation?
Which transformations commute?
CSE 681
Elementary Transformations
Theorem: Every affine transformation can be decomposed into elementary operations.
Elementary transformations:• Translation;• Rotation;• Scaling;• Shear.
Euler’s Theorem: Every rotation around the origin can be decomposed into a rotation around the x-axis followed by a rotation around the y-axis followed by a rotation around the z-axis. (or a single rotation about an arbitrary axis).
CSE 681
Vectors
Vector V = (Vx, Vy, Vz).
Homogeneous coordinates: (Vx, Vy, Vz, 0).
Affine transformation: .
010000
'
'
'
34333231
24232221
14131211
z
y
x
z
y
x
V
V
V
mmmm
mmmm
mmmm
V
V
V
CSE 681
Vectors
Rotation:
Scaling:
Translation:(Note: No change.)
.
01000
0100
00)cos()sin(
00)sin()cos(
0
'
'
'
z
y
x
z
y
x
V
V
V
V
V
V
.
01000
100
010
001
0
z
y
x
z
y
x
z
y
x
V
V
V
t
t
t
V
V
V
.
01000
000
000
000
0
'
'
'
z
y
x
z
y
x
z
y
x
V
V
V
s
s
s
V
V
V
CSE 681
Transform Parametric Line
P0
V
Q0
W
u
u
P(u)
M
Q(u)
MM
CSE 681
Line
Theorem: Affine transformations transform lines to lines.
transformed point on a line = point on transformed line
TTT VuMPMuVPM ][][][ 00
TT uVPMuPM ][)]([ 0
CSE 681
Affine Transformation of Lines
xz
y
xz
y
CSE 681
Affine Combinations
P
Q
P’
Q’
ab
b
a
R
M
R’
MM
CSE 681
Affine Combinations
Theorem: Affine transformations preserve affine combinations.
TTT bQMaPMbQaPM ][][][
TT bQaPMRM ][][
''][][][][ bQaPPaMPaMbQMaPM TTTT
CSE 681
Properties of Affine Transformations
• Affine transformations map lines to lines;• Affine transformations preserve affine combinations;• Affine transformations preserve parallelism;• Affine transformations change volume by | Det(M) |;• Any affine transformation can be decomposed into
elementary transformations.• Affine transformations [does/does not] preserve angles?• Affine transformations [does/does not] preserve the
intersection of two lines?• Affine transformations [does/does not] preserve distances?
CSE 681
Properties of Transformation Matrices
Second column is how (0,1,0,0) transforms
1000
0100
0010
0001
????
????
????
????
44434241
34333231
24232221
14131211
mmmm
mmmm
mmmm
mmmm
0
0
0
1
?
?
?
?
44434241
34333231
24232221
14131211
mmmm
mmmm
mmmm
mmmm
Matrix holds how coordinate axes transform and how origin transforms
Third column is how (0,0,1,0) transforms
First column is how (1,0,0,0) transforms
CSE 681
Properties of Pure Rotation Matrices
Rows (columns) are orthogonal to each other
A row dot product any other row = 0
Each row (column) dot product times itself = 1
RT = R-1
CSE 681
Coordinate Frame
• Coordinate frame is given by origin and three mutually orthogonal unit vectors, i, j, k - defined in (x,y,z) space
• Mutually orthogonal (dot products): i•j = ?; i•k = ?; j•k = ?.• Unit vectors (dot products): i•i = ?; j•j = ?; k•k = ?.
xz
y
ij
k
CSE 681
Orientation
Right handed coordinate system: Left handed coordinate system:
xz
y
(Out of page)
x
z
y
(Into page)
CSE 681
Orientation
xz
y
ij
k
Right handed coordinate system: Left handed coordinate system:
Cross product: i x j = ?
xz
y
ij
k
Cross product: i x j = ?
How do you test whether (i,j,k) is left handed or right handed?
CSE 681
Coordinate Transformations
xz
y
ij
k
Given object data points defined in the (i, j, k, ) coordinate frame,Given the definition of (i, j, k, ) in (x, y, z) coordinates,
How do you determine the coordinates of the object data points in the (x, y, z, 0) frame?
CSE 681
Coordinate change (Translation)
(x, y, z)
.
110001
c
b
a
z
y
x
xz
y
a
c
b
(0,0,0)
Change from (a,b,c,) coordinates to (x,y,z,0) coordinates:
1. Move (a,b,c, ) to (x,y,z,0) and invert2. Move data relative from (0,0,0) out to
position by adding
CSE 681
Coordinate change (Rotation)
.
110001
c
b
a
z
y
x
c
(x, y, z)
a
b
(0,0,0)
Change from (a,b,c,) coordinates to (x,y,z,0) coordinates:
y x
z
z-axis rotation by
1. Rotate (a,b,c) by and invert 2. Rotate data by -
CSE 681
Object Transformations
xz
y
Given (i,j,k,p) defined in the (x,y,z,0) coordinate frame, Transform points
defined in the (i, j, k, ) coordinate frame to the (x,y,z,0) coordinate frame.
Rotate object to get x to line up with i, then translate to
ij
k
.
10
z
y
x
1
0
0
0
0
0
1
0
0
1
0
ij
k
.
1000
0
0
0
zzz
yyy
xxx
kji
kji
kji
CSE 681
Object Transformations
xz
y
ij
k
ij
k
.
1000
zzzz
yyyy
xxxx
kji
kji
kji
Affine transformation matrix:
CSE 681
Coordinate Transformations
xz
y
ij
k
Given (i,j,k,) defined in the (x,y,z,0) coordinate frame, Transform points in (i,j,k,0) coordinate frame to (x,y,z, ) coordinate frame.
translate by - , then rotate to align i with x – then invert
CSE 681
Coordinate Transformations
1000
0
0
0
zyx
zyx
zyx
kkk
jjj
iii
xz
y
ij
k
R=
1000
100
010
001
z
y
x
T =
CSE 681
Coordinate Transformations
xz
y
ij
k
.
1000
zzzz
yyyy
xxxx
kji
kji
kji
Affine transformation matrix:
Apply RT to coordinate systemApply (RT)-1 to data = T-1R-1
CSE 681
Composition of coordinate change
z”0
x”
y”
M1 changes from coordinate frame (x,y,z,) to (x’,y’,z’,’).
M2 changes from coordinate frame (x’,y’,z’,’) to (x’’,y’’,z’’,’’).
Change from coordinate frame (x,y,z,) to (x’’,y’’,z’’,0): ?
z’
’ x’
y’
z
x
y
M2
M1
2
CSE 681
Composition of transformations - example
x
y
A
B
B’
CSE 681
Transformations of normal vectors
n is a unit normal vector to plane .
M is an affine transformation matrix.
How is n transformed, to keep it perpendicular to the plane, under:
• translation?
• rotation by ?
• uniform scaling by s?
• shearing or non-uniform scaling?
n
CSE 681
Non-uniform scale transformation
n is a unit normal vector to plane : (-nx,0)
M is a non-uniform scale transformation matrix that only modifies x-coordinates
If we just transform the n as a vector, then M nT = n’
But we want n’’ – that has a non-zero y-coordinate value
n n’ n
CSE 681
Transformations of normal vectorsPlanar equation: a x + b y + c z + d = 0.Let N = (a, b, c, d). (Note: (a, b, c) is normal vector)Let P = (x, y, z, 1) be a point in plane .Planar equation: N • P = 0.M is an affine transformation matrix. (MT is M transpose.)Let P’ = M PT.Find N’ such that N’ • P’ = 0 (transformed planar equation)N • P = 0;N PT = 0;N (M-1 M) PT = 0;(N M-1) (M PT) = 0;(N M-1) P’T = 0;So N’ = NM-1
To put in column-vector form, (N’)T = (NM-1)T = (M-1)TNT
So N’ = ((M-1)T NT)T and (M-1)T is the transformation matrix to take NT into N’T Note: If M is a rotation matrix, (M-1)T = M.
n P