Coordinate
transformations
Operations
over vectors (matrix form)
Scalar
(dot) product
Vector
(cross) product
Transformation of coordinates
Composition and inversion of
transformations
Operations over vectors (matrix form)
Vector is defined as a single column matrix
a =
Scalar (dot) product is
defined as
a*b = a1 b1 + a2 b2 + a3 b3,
or, in the matrix form,
a*b = aTb
Vector
(cross) product
is
a vector with components
a x b =
or in the vector form,
=Ab
Variety of coordinate systems.
World coordinate system.
Display coordinate
system (pixelated).
Object coordinate
system : object definition is not changed with respect to its own coordinate
system.
Many transformations are best specified in a relative fashion.
World->road->car->front
left wheel – scene graph. Going down the scene graph is postmultiplication of the current
transformation
matrix.
.
Each coordinate is represented by
the three x,y,z values.
r =
Right-handed
and left-handed coordinate systems.
We shall denote new coordinates after
transformation as
Translation
If the object is
translated without rotation, the relationship between the new and old point
coordinate is the following :
or
= 
+
Translation of coordinate axes is equivalent to translation of object in opposite direction.
First, let’s consider rotation in two dimensions
in plane x-y by angle 
. z-coordinate does not change.
Using
polar coordinates in plane Oxy, we see that polar coordinates of some point (
) in the result of rotation about the origin become (
). As 
and 
,
or
In the matrix form
If coordinate axes are rotated, the
object is rotated by angle -
and
A = B T
Rotation in 3 dimensions about arbitrary vector
U by angle 
is defined by the matrix
where 
- scaling coefficient
for x-coordinate, 
- for y-coordinate, 
- for z-coordinate.
Rotation, scaling can
be defined by the matrix multiply in the form
r’ = A r
Translation cannot be represented in this way.
To include
translations into matrix transformations, each vector is represented in homogeneous coordinates
R = 
Definition of
homogeneous coordinates :
P = wR,
where w is homogenous
parameter, varying from point to point. Components of vector P (xw,yw,zw,w)
are homogeneous coordinates of point (x,y,z). Cartesian coordinates x,y,z are
obtained from homogeneous coordinates by division
x = P1/P4
y = P2/P4
z = P3/P4
So, the representation
of point is not unique, like equation of line ax+by+c=0 has no unique
coefficients, only relations between coefficients are meaningful.
For translation,
honogeneous parameter is constant and equals to 1.
And all transformation matrices for rotation and scaling get the 4x4 dimensions :
and for translation looks like this :
Composition and
inversion of transformations
A series of transformations can be accumulated into a single
transformation matrix.
Suppose our object is defined as follows and we wish to place the object in our scene like this:
This can be accomplished in several ways, one of which is:
1. rotate object by -90 clockwise
the rotation matrix being
= 
2. translate object by vector (5,3,0)
the translation matrix being
The composition is
Pworld = 
Plocal = 
Plocal
- compositions
of transformations are non-commutative
e.g., translation(5,3,0) with subsequent rotation(z,-90) is not rotation(z,-90) with subsequent translation(5,3,0) - transformations
can be considered of as a change in coordinate system.
- transformations
matrices are multiplied together, giving a single composite transformation
matrix
- inverting
a transformation matrix produces the back (inverse of) transformation.
Pworld = M Plocal,
Plocal = M-1 Pworld
- premultiplication
effects a transformation in world or fixed coordinates.
M = M1
M2 M3 M4
... Mn
pre post
- postmultiplication
affects a transformation in local coodinates.
- Open
GL postmultiplies transformation, applies them in local coordinates.
Projection transformations
Mapping 3D object onto plane (screen). Number of
dimensions is decreased by one! Problems with the back transformation.
Parallel projection
Perspective projection
·
produces
foreshortening
·
non-linear
A
visual effect of non-linearity, distorted
contours of windows :
The
explanation lies in non-liear mapping of textures onto the screen:
Let's use the following example to construct
a matrix that performs a perspective
transformation:
From similar triangles, we can see
that y'/d = y/z
Thus, y' = yd/z , similarly, x' = xd/z , and z’ = d.
In homogeneous coordinates, in
matrix form,
= 
x’w’
= x
homogeneous
parameter w’ = z/d.