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Three Dimensional
Modeling
Transformations
Three Dimensional Modeling
Transformations

Methods for object modeling
transformation in three dimensions
are extended from two dimensional
methods by including consideration
for the z coordinate.
Three Dimensional Modeling
Transformations

Generalize from 2D by including z coordinate

Straightforward for translation and scale,
rotation more difficult

Homogeneous coordinates: 4 components

Transformation matrices: 4×4 elements
3D Point

We will consider points as column vectors.
Thus, a typical point with coordinates (x, y, z)
is represented as:
 x
 y
 
 z 
3D Point Homogenous Coordinate

A 3D point P is represented in
homogeneous coordinates by a 4-dim.
Vect:
 x
 y
P 
z
 
1 
3D Point Homogenous Coordinate

We don't lose anything

The main advantage: it is easier to
compose translation and rotation

Everything is matrix multiplication
 x
 y
 
z
 
1 

3D Coordinate Systems
Right Hand
 Left Hand coordinate
coordinate system:
system:

3D Transformation
In homogeneous coordinates, 3D
transformations are represented by 4×4
matrixes:
a b
d e

g h

0 0
c
f
i
0
t

t

t

1
x
y
z
3D Translation
Moving one object from
one point to other
point
3 D TRANSLATION
In translation, an object is displaced and direction
from its original position. If the displacement is given
by the vector
v= txi+txj+tzk,
The new object point p’=(x’,z’,z’) can be found by
applying the transform Ttx,ty,tz to p=(x,y,z)

Here tx=distance moved by object along x-axis
ty=distance moved by object along y-axis
tz=distance moved by object along z-axis
3D Translation

P is translated to P' by:
P  T  P
 x   1
 y   0
 
 z   0
  
 1  0
0 0 t x   x
1 0 t y   y 

0 1 tz  z
  
0 0 1  1 
Q1 find the new coordinates of a point p(4,2,1) after it is
translated by distance 2,3 and 5 units x,y and z-axis??
SOLUTION : here tx=2,tx3 and tz=5
x’
1 0 0 tx
y’
= 0 1 0 ty
z’
0 0 1 tz
1
0 0 0 1
x’
y’
z’
1
1
= 0
0
0
0
1
0
0
0
0
1
0
2
3
5
1
4
2
1
1
x’=4+2=6
y’=2+3=5
z’=1+5=6
hence the new coordinates will be p’(6,5,6)
3D Translation

An Object represented as a set of polygon surfaces, is translated
by translate each vertex of each surface and redraw the polygon
facets in the new position.
x, y, z 

T  t x , t y , t z 
x' , y' , z'
Inverse Translation:
T (t , t , t )  T (t ,t ,t )
1
x
y
z
x
y
z
T-tx,-ty,-tz
1
0
= 0
0
0
1
0
0
0 -tx
0 -ty
1 -tz
0 1
The transformed equation will become x’=x-tx
y’=y-ty
z’=z-tz
3D Rotation
3D Rotation

In general, rotations are specified by a
rotation axis and an angle. In
two-dimensions there is only one
choice of a rotation axis that leaves
points in the plane.
3D Rotation

The easiest rotation axes are those that parallel to
the coordinate axis.

Positive rotation angles produce counterclockwise
rotations about a coordinate axix, if we are looking
along the positive half of the axis toward the
coordinate origin.
Coordinate Axis
Rotations
Coordinate Axis Rotations

Z-axis rotation: rotation about z-axis in
anticlockwise direction
 x' cos 
 y '  sin 
 
 z'  0
  
1  0
 sin 
cos 
0
0
0
0
1
0
0  x 
0  y 

0  z 
  
1  1 
P  R z (θ)  P
ROTATION ABOUT Z-AXIS IN CLOCKWISEDIRECTION.
Note that the +ve values of rotation angle Ө will produce
a rotation in the anticlockwise direction whereas –ve
values ofӨ produce a rotation in the clockise direction.
 in this case angle Ө is taken as –ve. According to the
trigonometric law
cos(Ө)= cos Ө
sin(- Ө)= -sin Ө
Now the matrix become  x'  cos  sin  0 0  x 

 y '  sin 
 
 z'  0
  
1  0
cos 
0
0
P  R z ( )  P
0
1
0
0  y 


0 z
  
1  1 
Coordinate Axis Rotations

Obtain rotations around other axes through
cyclic permutation of coordinate parameters:
xyzx
X-axis rotation in anticlockwise direction:
0
 x' 1
 y ' 0 cos 
 
 z '  0 sin 
  
0
 1  0
0
 sin 
cos 
0
0  x 
0  y 

0  z 
  
1  1 
P  R ()  P
x
X-axis rotation in anticlockwise direction:
 x' 1
 y ' 0
 
 z '  0
  
 1  0
0
cos 
sin 
0
0
 sin 
cos 
0
0  x 
0  y 


0 z
  
1  1 
P  Rx( )  P
X-axis rotation in clockwise direction:
 In this case angle Ө is taken as –ve. According to the trigonometric
law
cos(Ө)= cos
sin(- Ө)= -sin Ө
P  Rx( )  P
 x' 1
 y ' 0
 
 z '  0
  
 1  0
0
cos 
 sin 
0
0
sin 
cos 
0
0  x 
0  y 

0  z 
  
1  1 
Y-axis rotation in anticlockwise direction
P  R ()  P
y
 x'  cos 
 y '  0
 
 z '   sin 
  
1  0
0 sin  0  x 
1 0 0  y 
 
0 cos  0  z 
  
0 0 1  1 
Y-axis rotation in clockwise direction
P  R y ( )  P
 x' cos 
 y '  0
 
 z '   sin 
  
1  0
0
1
0
0
 sin 
0
cos 
0
0  x 
0  y 

0  z 
  
1  1 
3D Scaling
SCALING TRANSFORMATION



A scaling transformation alters the size of an
object.
An object can be scaled(streched or shrunk)
along the x,y and z axis by multiplying all its
points by the scale factors sx,Sy,Sz
All points P(x,y,z) on the scaled shape will now
become P’(x’,y’,z’). Such that
x’=x.Sx
y’=y,Sy
z’=z.zSz
3D Scaling

Sx,Sy and Sz are scaling factors along x,y and z directions..
P’(x’y’z’)
Scaling
P(x,y,z
P  S  P
 x s x
 y   0
 
 z  0
  
1 0
0
0
sy
0
0
sz
0
0
0  x 
0  y 

0  z 
  
1  1 
SCALING TRANSFORMATION


If values of Sx,Sy and Sz<1 then size of
objects reduced or the object move closer to
the coordinate origin.
If values of Sx,Sy and Sz>1 then size of
objects increased or the object move farther
to the coordinate origin.
Quo. Magnify a prism by the vertices A(1,4,3),
B(4,2,3),(6,5,3) and D(3,4,5) to twice its size.
Solution:The new coordinates of prism is formed from
original coordinates as follow:
p’=S2,2,2.p
A’B’C’D’ =S2,2,2 . ABCD
2
= 0
0
0
2
= 8
6
1
0
2
0
0
8
4
6
1
0 0
0 0
2 0
0 1
12 6
10 8
6 10
1 1
1
4
3
1
4
2
3
1
6
5
3
1
3
4
5
1
Composite
3D Transformations
Composite Transformation

With the matrix representation , we can set up a matrix for any
sequence of transformations as a composite transformation
matrix by calculating the matrix product of the individual
transformations.

Forming products of transformation matrices is often referred
to as a concatenation, or composition, of matrices.

For column matrix representation of coordinate positions, we
form composite transformations by multiplying matrices in
order from right to left. That is, each successive
transformation matrix premultiplies the product of the
preceding transformation matrices.
Composite 3D Transformations
Same way as in two dimensions:


Multiply matrices

Rightmost term in matrix product is the first
transformation to be applied
SCALING ABOUT THE FIXED POINT

Scaling with respect to a selected fixed position(x,y,z)
can be represented with the following transformation
equation sequence.
1. translate the fixed point to the origin.
2. Scale the object relative to the coordinate origin.
3.translate the fixed point back to its original position.
The matrix representation for an arbitrary fixed-point scaling can
then be expressed as the concatenation of these translate-scaletranslate transformation as
P’=T(xf,yf,zf).S(sx,sy,sz).T(-xf,-yf,-zf).P
X’ 1
Y’ = 0
Z’ 0
1 0
0
1
0
0
0
0
1
0
tx
ty
tz
1
 x'   s x
 y '  0
 
 z'   0
  
1   0
sx 0
0 sy
0 0
0 0
0 0 1 0
0 0 0 1
sz 0 0 0
0 1 0 0
0
0
sy
0
0
sz
0
0
0
0
1
0
tx
ty
tz
1
(1  s x ) x f   x 
 y
(1  s y ) y f 
 
(1  s z ) z f   z 
 
1
 1 
x’ x
y’ y
z’ z
1 1
3D Reflections
3D Reflections

About an axis: equivalent to
180˚rotation about that axis
3D Reflections
About a plane:

A reflection through the xy plane:
 x  1
 y  0
 
  z  0
  
 1  0

0  x 
1 0 0  y 
 
0  1 0  z 
  
0 0 1  1 
0
0
A reflections through the xz and the yz
planes are defined similarly.
3D Shearing
3D Shearing


Modify object shapes
Useful for perspective projections:
 E.g. draw a cube (3D) on a screen (2D)
 Alter the values for x and y by an amount
proportional to the distance from zref
3D Shearing
1
0
M zshear  
0

0
0 shzx
1 shzy
0
1
0
0
 shzx  z ref 
 shzy  z ref 

0

1

Transformations
Between
Coordinate Systems
Transformations Between Coordinate ystems


Translate such that the origins overlap
Rotate such that the axes overlap
y
y'
uy '
( x0 , y0 , z0 )
(0,0,0)
x
x'
uz '
z'
z
u ' x1
u '
R  T   y1
 u ' z1

 0
ux '
u'x 2
u'x3
u' y 2
u' y3
u'z 2
u'z3
0
0
0
0
 T  x0 , y0 , z0 
0

1
Tilting transformation



Tilting is defined as rotation about X-axis followed by
rotation about Y-axis or vice versa. So it is a composite
transformation.
The effect of first transformation i.e rotation about X-axis
will make the given vector lie in the YZ plane.
The effect of second transformation rotation about y-axis
will make the vector aligns the given line either with Zaxis or with the x=axis
In order to obtain the required tilting matrix,
we have to perform the following two
rotations
1.rotation about x-axis by an angle α
2.rotation about y-axis by an angle β

y
y
Rx(β)
Rx(a)
x
x
z
z