07 Coordinate System Change

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Universidade Federal da Paraíba
Centro de Informática
Change of Coordinate System
Lecture 7
1107190 - Introdução à Computação Gráfica – Turma 01
Prof. Christian Azambuja Pagot
CI / UFPB
Universidade Federal da Paraíba
Centro de Informática
Why do we need to change 
basis?
● Sometimes it is necessary to transform the 
coordinates of a primitive to a new coordinate 
system.
● This is the case, for instance, when we want to 
transform vertices from the Universe Space 
to the Camera Space.
Universidade Federal da Paraíba
Centro de Informática
Universe vs. Camera Space
z
y
Universe
Space
z
y
Camera
Space
z
u
y
u
y
c
z
c
Universidade Federal da Paraíba
Centro de Informática
Universe vs. Camera Space
z
y
Camera
Space
y
c
x
c
Universidade Federal da Paraíba
Centro de Informática
Universe vs. Camera Space
z
y
Camera
Space
y
c
x
c
Universidade Federal da Paraíba
Centro de Informática
Coordinate System Basis
● Linear independence.
● Orthogonality.
● Orthonormal basis.
e
1
e
2
e
3
e1=[100] ,e2=[010 ] ,e3=[001] [e1 e2 e3 ]=[1 0 00 1 00 0 1]
Universidade Federal da Paraíba
Centro de Informática
Arbitrary Vectors
● An arbitrary vector w:
w
e
1
e
2
e
3
α
1
α
2
α
3
w= [α1 e1+α2e2+α3 e3 ]
a=[α1α2α3]
w= [e1 e2 e3 ]a
w=a
[1 0 00 1 00 0 1]
Universidade Federal da Paraíba
Centro de Informática
Change of Basis
● An arbitrary vector w:
w
e
1
e
2
e
3
v
u
n
Universidade Federal da Paraíba
Centro de Informática
Change of Basis
● An arbitrary vector w:
w
β
2
β
3
β
1
v
u
n
w= [β1u+β2 v+β3 n ]
b=[β1β2β3 ]
w= [u v n ]b
Universidade Federal da Paraíba
Centro de Informática
e
1
e
2
e
3
α
1
α
2
α
3
Change of Basis
● An arbitrary vector w:
ww
β
2
β
3
β
1
v
u
n
w= [β1u+β2 v+β3 n ]
b=[β1β2β3 ]
w= [u v n ]b
w= [e1 e2 e3 ]a w=a
a=Bb
B=[u v n ]
b=B−1a
b=BT a
B is orthogonal
(contains only rotations)!
Its inverse is its transpose!
Universidade Federal da Paraíba
Centro de Informática
e
1
e
2
e
3
Change of Basis
● How about two basis such that the origins do 
not match? v
u
n
t=[ t xt yt z ]
Universidade Federal da Paraíba
Centro de Informática
Change of Basis
● How about two basis such that the origins do 
not match?
– 1st: Match both space origins through 
translation!
– 2nd: Apply the rotation matrix, as before.
b=BT a '
a '=a−t
Can we pack all transformations
into a unique matrix?
Universidade Federal da Paraíba
Centro de Informática
Change of Basis
● We will have to use homogeneous 
coordinates due translation:
BT=[ux uy uz 0vx v y vz 0nx ny nz 0
0 0 0 1
] T=[1 0 0 −t x0 1 0 −t y0 0 1 −t z
0 0 0 1
]
M=BTT=[ux uy uz −t⋅uv x vy vz −t⋅vnx ny nz −t⋅n
0 0 0 1
] b=Ma
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