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Transformations
Bed
ř
ich Beneš, Ph.D.
Purdue University
Department of Computer Graphics
© Bedrich Benes
Transformations
•
transformations are essential
•
closely related to Cg programs (vp)
•
the following should be
a
review
of wellknown things (is it?)
© Bedrich Benes
Homogenous coordinates
•
We use
homogenous coordinates
in 3D
A=[x,y,z,w]
•
The relation between homogenous and
Cartesian coordinates is
[x,y,z]=[x/w,y/w,z/w]
•
for w=0 the [x,y,z,w]
corresponds to a vector (x,y,z)
© Bedrich Benes
Homogenous coordinates
•
Advantages of using homog. coords:
•
Linear transformations (scale, shear,
translate, rotate) can be represented by
matrices 4x4
•
Projective transformation can be represented
by matrices 4x4
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Transformations
•
Having matrix
A
of type 4x4
and a vertex v=[x,y,z,w]
The vertex transformation is represented
by the matrix multiplication
v’=
A
v
•
to transform a vertex only the matrix must
be specified
© Bedrich Benes
Transformations
•
Example:
•
A
corresponds to the translation
by the vector (X
T
,Y
T
,Z
T
)
A
=
1 0 0 X
T
0 1 0 Y
T
0 0 1 Z
T
0 0 0 1
© Bedrich Benes
Transformations
•
One of the first things ever performed by
hardware is matrixtovector multiplication
•
Every vertex must pass a various stages
of transformations
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 Spring '08
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