Review of Vectors
•
Right-handed versus left-handed systems
•
Positive rotation angles for right-handed systems
-2•
Homogeneous coordinates of a 3D point
- Idea: add a third coordinate: (x, y, z) --> (x h , y h , z h , w)
- Homogenize (x h , y h , z h , w):
x=
yh
zh
xh
, y=
,z= ,w≠0
w
w
w
- In general: (x, y, z) --> (xw, yw, zw, w) (i.e., x h =xw, y h =yw, z h =zw)
- w can assume any value (w ≠ 0), for example, w = 1:
(x, y, z) − − > (x, y, z, 1) (no division is required when you homogenize !!)
(x, y, z) − − > (2x, 2y, 2z, 2) (division is required when you homogenize !!)
- Each point (x, y, z) correponds to a line in the 4D-space of homogeneous
coordinates
-3-
3D Geometrical Transformations
•
Translation
y
.
P’(x’,y’,z’)
.
P(x,y,z)
x
z
 x′   1
 y′   0
 =
 z′   0
 1  0
0
1
0
0
0
0
1
0
dx   x 
dy   y 
 
dz   z 
1  1
(Verification: x′ = 1x + 0y + 0z + 1dx = x + dx)
(Verification: y′ = 0x + 1y + 0z + 1dy = y + dy)
(Verification: z′ = 0x + 0y + 1z + 1dz = z + dz)
(Homogenize: divide by 1 !!)
P′ = T (dx, dy, dz) P
•
Scale
-4 x′   s x
 y′   0
 =
 z′   0
1 0
0
sy
0
0
0
0
sz
0
0  x 
0  y
 
0  z 
1 1
P′ = S(s x , s y , s z ) P
•
Rotation
- Rotation about the z-axis
x′ = xcos( ) − ysin( )
y′ = xsin( ) + ycos( )
z′ = z
 x′   cos( ) −sin( )
 y′   sin( ) cos( )
 =
0
 z′   0
0
1  0
P′ = R z ( ) P
0
0
1
0
0  x 
0  y
 
0  z 
1 1
-5- Rotation about the x-axis
x′ = x
y′ = ycos( ) − zsin( )
z′ = ysin( ) + zcos( )
 x′   1
 y′   0
 =
 z′   0
 1  0
0
0
cos( ) −sin( )
sin( ) cos( )
0
0
P′ = R x ( ) P
0  x 
0  y
 
0  z 
1 1
-6- Rotation about the y-axis
x′ = zsin( ) + xcos( )
y′ = y
z′ = zcos( ) − xsin( )
 x′   cos( )
 y′   0
 =
 z′   −sin( )
1  0
sin( )
0
cos( )
0
0
1
0
0
P′ = R y ( ) P
0  x 
0  y
 
0  z 
1 1
-7•
Change of coordinate systems
- Suppose that you know the coordinates of P 3 in the xyz system and that you
need its coordinates in the R x R y R z system.
* You need to recover the transformation T from R x R y R z to xyz.
* Apply T on P 3 to compute its coordinates in the R x R y R z system.
- Assume that u x , u y , and u z are the unit vectors in the xyz coordinate system.
- Assume that r x , r y , and r z are the unit vectors in the R x R y R z coordinate system (important: r x , r y , and r z are represented in the xyz coordinate system).
- Find a mapping that will map r z → u z , r y → u x , and r y → u y
 0   a11
 0   a21
 =
 1   a31
1  0
a12
a22
a32
0
a13
a23
a33
0
0   rzx 
0   rzy 
   (r z → u z )
0   r zz 
1  1 
(1)
 1   a11
 0   a21
 =
 0   a31
1  0
a12
a22
a32
0
a13
a23
a33
0
0   r xx 
0   r xy 
 (r x → u x )

0   r xz 
1  1 
(2)
-8 0   a11
 1   a21
 =
 0   a31
1  0
a12
a22
a32
0
a13
a23
a33
0
0   r yx 
0   r yy 
 (r y → u y )

0   r yz 
1  1 
From (1): aT3 r z = 1 or a3 = r z
From (2): aT1 r x = 1 or a1 = r x
From (3): aT2 r y = 1 or a2 = r y


Thus, R = 


0
r Tx
r Ty
r Tz
0
- Verify
  1
 rx   0 
R  =  
  0
 1  1
  0
 ry   1 
R  =  
  0
 1  1
  0
 rz   0 
R  =  
  1
 1  1
0
0

0
0

1
(3)